TECHNICAL MATHEMATICS
ALGEBRA,
GEOMETRY,
TRIGONOMETRY
Prepared by
Rabie Soliman:
Senior Teacher of Math (Sur V T C)
Ridha Bechir Gharbi: Senior Teacher of Math (Sur V T C)
Mohammed Fatoh:
Senior Teacher of Math (Saham V T C)
Mahmod Badrawy:
Senior Teacher of Math (Saham V T C)
Deogracias E. Joaquin:
Senior Teacher of Math (Shinas V T C)
Raafat Sayed Mohammed:
Revised by
Senior Teacher of Math (Saham V T C)
Ghareeb Zaki
:
Curriculum Specialist of Mathematics & Physics
2
3
‫مكدمــــــــــــ٘‬
‫بضه اهلل الزمحً الزسٔه‬
‫( ّقل رب سدىٕ علناَ )‬
‫صدق اهلل العظٔه‬
‫قاو بإعداد املادٗ العلنٔ٘ هلذا اللتاب الشمالء املعلنٌْ مً مزاكش التدرٓب املَين‪ -‬صْر‪ ،‬صشه‪ ،‬عياص ‪ّ -‬قد قنيا مبزادع٘‬
‫تلم املادٗ العلنٔ٘ ّتصئفَا بالغلل املكبْل ّاملياصب كنزدع جلنٔع الشمالء معلنٕ الزٓاضٔات مبزاكش التدرٓب املَين‬
‫احللْمٔ٘ ّمعَدٖ تأٍٔل الصٔادًٓ‪ ،‬سٔح قنيا بتصئف املادٗ العلنٔ٘ علٕ ٍٔئ٘ ثالخ ّسدات ٍٕ اجلرب ّاهليدص٘ ّسضاب‬
‫املجلجات‪.‬‬
‫ٓغتنل اجلشء اخلاص باجلرب علٕ صت٘ فصْل تغطٕ معظه مْضْعات امليَر للنضتْٓني األّل ّالجاىٕ مبزاكش التدرٓب املَين‬
‫ّمعاٍد تأٍٔل الصٔادًٓ ّفكا للنياٍر املطْرٗ‪ّٓ ،‬غتنل اجلشء اخلاص باهليدص٘ عل‪ ٙ‬فصلني ٓغطٔاٌ معظه مْضْعات‬
‫اهليدص٘ املضتْٓ٘ ّالفزاغٔ٘(ٍيدص٘ الفطاء) ّاهليدص٘ التشلٔلٔ٘(اإلسداثٔ٘) املكزرٗ للنضتْٓني األّل ّالجاىٕ مبزاكش التدرٓب‬
‫املَين ّفكا للنياٍر املطْرٗ‪ ،‬أما اجلشء اخلاص حبضاب املجلجات فٔغتنل عل‪ ٙ‬فصلني ٓغطٔاٌ معظه مْضْعات املياٍر املكزرٗ عل‪ٙ‬‬
‫املضتْٓني األّل ّالجاىٕ مبزاكش التدرٓب املَين‪.‬‬
‫ىغلز مجٔع الشمالء معلنٕ الزٓاضٔات الذًٓ صاٍنْا بإعداد املادٗ العلنٔ٘ ‪ :‬أ‪ .‬ربٔع صلٔناٌ مً مزكش صْر عل‪ ٙ‬دلَْدِ‬
‫الْافز يف إعداد املادٗ العلنٔ٘ باإلضاف٘ إىل قامْظ املصطلشات الزٓاضٔ٘ ّالذٖ قنيا مبزادعتُ ّسذف ّإضاف٘ بعض الللنات ّقد‬
‫ّضعياِ يف مؤخزٗ املزدع كٕ ٓلٌْ معٔيا للشمالء املعلنني أٓطا أبيائيا املتدربني يف مزاكش التدرٓب املَين ّمعَدٖ تأٍٔل‬
‫الصٔادًٓ‪ ،‬كنا أتْدُ بالغلز إىل أ‪ .‬رضا البغري مً مزكش صْر عل‪ ٙ‬املادٗ العلنٔ٘ اليت كاىت ميظن٘ ّمزتب٘ بغلل دٔد ّمل‬
‫تلً صٔغ الكْاىني اليت اصتددمَا بعٔدٗ عً تلم املضتددم٘ ف‪ ٙ‬التعلٔه العاو بضلطي٘ عناٌ ‪.‬‬
‫كنا أتْدُ بالغلز إىل أ‪ .‬ذلند فتْح مً مزكش صشه عل‪ ٙ‬دلَْدِ الْفري يف إعداد املادٗ العلنٔ٘ اخلاص٘ حبضاب املجلجات ‪،‬‬
‫كنا أتْدُ بالغلز إلٕ أ‪ .‬ذلنْد بدراّٖ مً مزكش صشه عل‪ ٙ‬املادٗ العلنٔ٘ الغشٓزٗ ّاخلاص٘ باهليدص٘ املضتْٓ٘ ّالفزاغٔ٘‪،‬‬
‫كنا أتْدُ بالغلز اىل ك ًال مً أ‪ .‬رأفت صٔد مً مزكش صشه ‪ ،‬أ‪ .‬دٓٔذْراصٔط مً مزكش عياص عل‪ ٙ‬مضاٍنتَه يف إعداد‬
‫املادٗ العلنٔ٘ ‪.‬‬
‫كنا أتكدو بالغلز إىل الفاضل‪ /‬منز محاد – أخصائٕ ّصائل تعلٔنٔ٘ بدائزٗ تطْٓز املياٍر ‪ -‬عل‪ ٙ‬تعاّىُ يف إخزاز ٍذا املزدع‬
‫بالغلل املكبْل‪.‬‬
‫ّأخريا أردْ أٌ ٓلٌْ ٍذا املزدع عْىا إلخْاىيا املعلنٌْ عل‪ ٙ‬أداء رصالتَه الرتبْٓ٘ ّالعلنٔ٘ ّفكا مليظْم٘ التدرٓب املَين يف‬
‫صلطي٘ عناٌ‪.‬‬
‫( ربيا ال تؤاخذىا إٌ ىضٔيا أّ أخطاىا )‬
‫غريب زكي‬
‫أخصائٕ مياٍر الزٓاضٔات ّالفٔشٓاء‬
‫مضكط ‪2010-‬‬
‫‪4‬‬
‫و‬
Contents
Chapter No.
0
Subject
Page No.
INTRODUCTION
4
NUMERICAL SYSTEMS
8
1.1
Sets of numbers
8
1.2
Operations with decimals
11
1.3
Operations with fractions
14
1.4
Percentage and ratio
20
1.5
Approximation
21
1.6
Ratio
22
1.7
Proportion
22
1.8
Simple interest
23
1.9
Mental arithmetic
24
1
2
POLYNOMIALS
27
2.1 Operation on polynomials
27
2.2 Factorizing polynomials
30
SOLUTION OF EQUATIONS AND INEQUALITIES
35
3.1 Solution of linear equations
35
3.2 Solution of linear inequality in one variable
36
3.3 Solving Quadratic Equation
37
SEQUENCES AND SERIES
40
4.1 Sequences
40
4.2 Series
41
4.3 Arithmetic Sequences and Series
42
4.4 Geometric Sequences and Series
45
LOGARITHMS
49
5.1 Exponential and Logarithmic Functions
49
PERMUTATION, COMBINATIONS AND BINOMIAL
57
6.1 Introduction
57
6.2 Permutations
58
6.3 Combinations
58
6.4 Binomial Expansion
59
3
4
5
6
5
7
PLANE AND SOLID GEOMETRY
61
7.1 Area of Squares and Rectangles
61
7.2 Area of Triangles
63
7.3 Area of Parallelograms
66
7.4 Area of Trapezoids
68
7.5 Circumference and Area of Circles
70
7.6 Surface Areas of Prisms and Cylinders
72
7.7 Surface Area of Pyramids and Cones
75
7.8 Volume of Prisms and Cylinders
79
7.9 Volumes of Pyramids and Cones
81
7.10 Surface Area and Volume of Spheres
83
7.11 Table of Area
89
7.12 Table of Solids
91
ANALYTIC GEOMETRY
94
8.1 Cartesian Coordinate
94
8.2 Mid-point Coordinate of a Line Segment
95
8.3 Slope of a Straight Line
95
8.4 Straight Line Equation
97
8.5 Parallel and Perpendicular Lines
97
8
TRIGONOMETRIC FUNCTIONS
103
9.1 The Pythagorean Theorem
103
9.2 Degree Measure
111
9.3 Radian Measure
113
9.4 The Trigonometric Functions of Acute Angles
118
9.5 Trigonometric Functions of Special Angles
126
SOLUTION OF TRIANGLE
132
10.1 Sine Rule
132
10.2 Cosine Rule
137
10.3 Solving Right Triangles
140
10.4 Solving Oblique Triangles
143
10.5 Applications Of Solving Triangles
151
DICTIONARY
164
9
10
APPENDIX
6
ALGEBRA
Chapter 1: Numerical Systems
Rabie Soliman (Sur V T C)
Chapter 2: Polynomials
Deogracias E. Joaquin (Shinas V T C)
Chapter 3: Solution of Equations and Inequalities
Raafat Sayed Mohammed (Saham V T C)
Chapter 4: Sequences and Series
Rabie Soliman (Sur V T C)
Ridha Bechir Gharbi (Sur V T C)
Chapter 5: Logarithms
Raafat Sayed Mohammed (Saham V T C)
Ridha Bechir Gharbi (Sur V T C)
Chapter 6: Permutation, Combinations and
Binomial theorem
Ridha Bechir Gharbi (Sur V T C)
7
Chapter 1
Numerical Systems
Sets of numbers
The set of natural numbers N
Where
N  0,1,2,3,............
The set of Integers z
Where
Z  ...........,3,2,1,0,1,2,3,.......
We can divide this set into
The set of Positive integers
Z    1,2,3,......
the set of Negative integers
Z    1,2,3,......
The set of non - zero integers
Z *  Z  0
Note that
Z  Z   0 Z 
Z  N  Z
8
The set of rational numbers Q
Where
p

Q   : p, q  Z , q  0
q

/
The set of Irrational numbers Q
This set contains the numbers which cannot put on the form
p
, q  0 , such as 5 , 3 9 ,.......
q
Note that
Q n Q'= Ø
DRILL 1
Which of the following numbers belongs to
Q and which of them belongs to Q /
25
,
9
3
27 ,
7 , 2.1 , 0.4
9
The set of real numbers
The set of real numbers is the union of the set of rational numbers and the set of irrational numbers.
Let R denotes the set of real numbers so
R  Q Q/
Real numbers
R
Rational numbers
Irrational numbers
Q
Q/
Non integers
Integers
Z
Negative integers
Natural numbers
Z
N
0 
Positive integers
Z
10
Operations with decimals
Addition of decimals
When adding decimal quantities place the numbers in columns so that the decimal points occur directly
underneath one another.
Eg.1
Find the value of:
3.518  1.64  5.047
Solution
3. 5 1 8
1. 6 4
5. 0 4 7
10. 2 0 5
Subtraction of decimals
When subtracting decimal quantities use the same column arrangement as for addition taking care to
write the decimal points directly underneath one another.
Eg.2
Find the value of 14.301-8.576
Solution
14. 3 0 1
8 .57 6
__________
5.725
11
Multiplication of decimals
When multiplying two decimal quantities the number of decimal places in the answer will be the same as
the total number of decimal places in both quantities.
Eg.3
Find the value of 2.78 × 1.3
Solution
Step 1
Disregard the decimal points
2 7 8
1 3
_________
8 3 4
2 7 8 0
__________
3 6 1 4
Step 2
The total number of decimal places in both numbers is 2+1=3.
Step 3
Insert the decimal point. Then the answer is 3.614
Drill 1
Multiply 5.62 by 3.13
Division of decimals
To make division by a decimal quantity the divisor must be converted to a whole number by multiplying
by 10, 100, 1000, etc.
The dividend must also be multiplied by the same value.
12
Eg.4
Find the value of 33.5 divided by 1.34
Solution
Take care 33.5 is the dividend
1.34 is the divisor
To find the answer do as follows:Step 1
Convert the divisor to a whole number by multiply it by 100
1.34 × 100=134
Step 2
Multiply the dividend by the same value
33.5 × 100=3350
Step 3
Divide 3350 by 134
25
1343350
2 6 8
6 7 0
6 7 0
0 0 0
The answer is 25
13
Operations with fractions
Vulgar fractions
This type of fraction has a value which is always less than 1.The number above the dividing line called the
numerator is always less than the number below the dividing line which is called the denominator.
Eg.1
1 3 2 6
  
7 7 7 7
(Note: the fractions have a common denominator so we add the numerators)
Eg.2
11 5
6
 
13 13 13
Improper fractions
In this fractions the numerator is greater than the denominator such as:
9 17 5
, ,
5 8 2
Mixed fractions
If the numerator is greater than the denominator then the answer is a whole number and a vulgar fraction
and this is called a mixed number or mixed fraction thus:
7
2 2 1 3
 1 ,1 , 3 , 7
5
5 3 2 5
Are mixed numbers.
Note that
Mixed numbers can be converted into improper fractions by multiplying the whole number by the
denominator and adding the numerator.
Eg.3
4 3  5  4 19
3 

5
5
5
14
Drill 2
Place each of the following values under head in the following table
5
2 12
6 13
3
,3 ,
,4 ,
,2
7
5 17
7 10
7
Vulgar fraction
Improper fraction
Mixed number
Addition of fractions
1- If the fractions have the same denominator (common denominator) we add the numerator.
Eg.4
3 2 5
 
7 7 7
2- If the fractions have different denominators
In this case we replace the fractions by an equivalent fractions to give a common denominator.
Eg.5
Simplify
1 2

4 3
Solution
Both fractions must be replaced by equivalent fractions
1 1 3 3


4 4  3 12
2 2 4 8


3 3  4 12
Then
1 2 3 8 11
   
4 3 12 12 12
Eg.6 Find the value of:
1 3 5
3 1 1
2 4 8
15
Solution
There are two methods:
1- Collect the whole number together
1 3 5
4  1  2  3  1 3
3 1 1  5 
2 4 8
8
465
 5
8
15
 5
8
7
6
8
2-converting the mixed numbers to improper fractions
1 3 5 7 7 13
3 1 1   
2 4 8 2 4 8
4  7  2  7  113

8
28  14  13 55
7


6
8
8
8
Subtraction of fractions
Eg.7
Simplify
5 3

6 4
Solution
5 3 2  5  3  3 10  9 1
 


6 4
12
12
12
Drill 3
Simplify
2
5
5
4  2 3
5
7 13
16
Multiplication of fractions
To carry out multiplication we multiply the numerators together and the denominators together.
Eg.8
Simplify
3 2

7 3
Solution
3 2 3 2 6 3
 


7 3 7  3 21 7
Drill 4
Simplify
3 1 3
1  
5 4 7
2 1
2 1  3
3 2
Division of fractions
Eg.9
Simplify
4 2

5 3
Solution
4 2 4 3 6
1
    1
5 3 5 2 5
5
17
Conversion of fractions to decimals
Eg. 10 Convert to decimal
2
5
7
2)
8
1)
Solution
1)
2
 25
5
0.4
5 2.0
Then
2)
2
 0.4
5
7
 7 8
8
8
Then
0.875
7.000
64
_____
60
56
_____
40
40
_____
00
7
 0.875
8
18
Conversion of decimals to fractions
In a decimal fraction the first figure to the right of the point gives the number of tenths; the second figure
gives hundredths, the third gives thousandths.
Eg.11 For the decimal 0.531
Place the values
1
ths
10
0.5
1
ths
100
3
5
Thus 0.5 
10
531
i.e 0.531 
1000
1
ths
1000
1
0.53 
1
ths
10000
53
100
0.001 
1
1000
Eg.12 Convert 9.345 to a mixed number
Solution
The whole number 9 does not change
0.345 
345
1000
Then 9.345  9
345
1000
19
Percentage and ratio
Conversion of fractions to percentages
To convert a fraction ( or decimal fraction (To a percentage, multiply by 100.
.Eg.13
Convert
3
to a percentage
20
Solution
3
 100  15%
20
Eg.14
Convert 0.35 to a percentage
Solution
0.35×100=35%
Conversion of percentages to fractions
To convert a percentage to a fraction divide by 100.
Eg.15
Convert 42% to a vu lg ar fraction
Solution
42% 
42 21

100 50
Eg.16 Convert 14 % to decimal fraction
Solution
14
14% 
 0.14
100
Percentage of quantities
The percentage of quantity can be found by multiplying the quantity by the fraction equivalent of the
percentage.
Eg.17 Find 12% of 300 kg
Solution
12
 300  36kg
100
20
Approximation
To reduce the number of decimal places does as follows:
1- If the first figure after the required number of places is 5 or greater, then add 1 to the previous figure
and omit the first figure.
2- If the first figure after the required number of places is less than 5 then simply omit the first figure
without any change.
Eg.5
2.345 correct to 2 decimal places is 2.35
4.093 correct to 2 decimal places is 4.09
Exercises
I) find the value of)
1- 43.3+9.15+10.06
2-3.8451+0.219+11.2713
3-2.103+0.125+1.3518+0.0073
4-87.64-37.95
5-40.589-18.335
6-17.408-9.075
7-8.305×4.64
8-1.5×2.6×0.3
9-135.5×14.3
10-5.58÷1.8
11-592.47÷9
12-62.3÷0.7
(ii) Multiply 4.71 by 2.35 and give the answer correct to 2 decimal places
(iii) Show the value 1.8919 correct to:
1- 3 decimal places.
2- 2 decimal places.
21
Ratio
The relationship between two quantities having the same units may be expressed in the form of a ratio.
Eg.18 Consider two metal bars A and B where A is 5m long and B is 7m long. The length of A is to.
5 : 7 or
5
The length of B as
7
Note that
5 is called the first term, 7 is called the second term and both 5, 7 called the terms of the fraction.
Proportion
Definition
Proportion is an equality of two or more ratios (equivalent fractions)
Eg .
1 3 12
 
 ............
2 6 24
Drill 5
Complete the following table to make the corresponding numbers in two rows proportional
…..
15
75
…..
…..
…..
2
10
50
…..
…..
…..
22
Important remark
* We know that:
2 6

3 9
2 is called the first term.
3 is called the second term.
6 is called the third term.
9 is called the fourth term.
Both 2, 9 called the extremes.
Both 3, 6 called the means.
* The product of the extremes = the product of the means.
Drill 6
Find the value of x if
5 x

3 2
Drill 7
The price of 20 liters of oil is RO 10 . Evaluate:
1-the price of 30 liters of the same oil.
2-number of liters of price RO 12.5.
Simple interest
I
CPT
100
WHERE
I is the simple int erest
C principal amount
P rate of int erest
T time int erval in years
Drill 8
Salem borrowed OR 30000 from bank Muscat to open a project if the rate of interest charged is 8% per
annum. Find the interest at the end of 5 years.
23
Mental arithmetic
Divisibility by 2
Any number is divisible by 2 if its units digit contains 0, 2,4,6,8. Such as 234,1026,18 etc.
Divisibility by 3
Any number is divisible by 3 if the sum of its digits is divisible by 3 such as 2001, 429, 9342, etc.
Divisibility by 4
Any number is divisible by 4 if the number formed by the units digit and tens digit is divisible by 4 such as
100, 416, 2324, etc.
Divisibility by 5
Any number is divisible by 5 if its unit's digit contains 0 or 5.such as 500, 515, 7015, etc.
Divisibility by 6
Any number is divisible by 6 if it is divisible by both 2and 3 such as 5310, 300, 150, etc.
Divisibility by 9
Any number is divisible by 9 if the sum of its digits is divisible by 9 such as 513, 402030, etc.
Divisibility by 11
Any number is divisible by 11 if the difference between the sum of the odd places and the sum of the even
places is divisible by 11 or the difference is 0 such as 10846 where :
The sum of the odd places =6+8+1=15
The sum of the even places=4+0=4
The difference =15-4=11
So 10846 are divisible by 11.
Also 1331 is divisible by 11.
Divisibility by 25
Any number is divisible by 25 if the number formed by the digits in the units place and tens place is
divisible by 25 such as 100, 2125, etc.
Multiplying by 9
To multiply a number by 9 multiply this number by 10 and subtract the original number from the result.
24
Eg.1 Find 46×9
Solution
46×9=46×10-46=414
Multiplying by 99
To multiply a number by 99 multiply this number by 100 and subtract the original number from the result.
Eg.2 Find 46×99
Solution
46×99=46×100-46=4554
Multiplying by 999
To multiply a number by 999 multiply this number by 1000 and subtract the original number from the
result.
Eg.3 Find 46×999
Solution
46×999=46×1000-46=45954
Multiplying by 11
To multiply a number by 11 multiply this number by 10 and add the original number to the result.
Eg.4 Find 2643×11
Solution
2643×11= 2643×10+2643=29073
Multiplying by 5
To multiply a number by 5 add 0 for the number and divide the result (new number) by 2
Eg.5 Find 4286×5
Solution
4286×5=42860÷2=21430
25
Multiplying by 25
To multiply number by 25 add 00 to the number and divide the result (new number) by 4
Eg.6 Find 58×25
Solution
58×25=5800÷4=1450
Multiplying by 125
To multiply number by 125 add 000 to the number and divide the result (new number) by 8
Eg.7 Find 48×125
Solution
48×125=48000÷8=6000
Multiplying by 15
To multiply number by 15 we add the number with its half and multiply the result by 10.
Eg.8 Evaluate 15×24
Solution
24+12=36
36×10=360
Then
15×24=360
26
Chapter 2
Polynomials
Pre-requisite knowledge:




(2-1)

Operations of integers
Laws of exponents
Identify similar terms
Identify numerical and literal coefficients
Operations on Polynomials
In adding polynomials, combine similar terms.
Examples: Find the sum.
1)  3 x
2)  7 y 2
3)  12 x 3 y
 2x
 3y 2
 5x 3 y
 5x
 10 y 2
 7x3 y
4) 7 a  3b
5a  b
5) 15 x  10 y  5
9x  6 y  3
12a  2b

6) 7 x 2  2 x  10
 5x  3
24 x  4 y  2
7 x 2  3 x  13
In subtracting polynomials, change the sign of all terms of the subtrahend and proceed to
the rule of addition.
Examples: Find the difference.
1)  5 x
 7x
( subtrahend )

 5x
 7x
(add )
 2x
2) 6a  3b
5a  2b
( subtrahend )

6a  3b
 5a  2b
a  5b
27
(add )
3) x 3  2 x 2  x  7
5 x 2  3x  2
x3  2x 2  x  7
( subtrahend )

 5 x 2  3x  2
x 3  3x 2  4 x  9

Removal of a Grouping Symbol
A grouping symbol can be removed by observing these rules:
1. If the grouping symbol is preceded by a plus sign (+), remove the plus
sign and the grouping symbol right away.
2.
If the grouping symbol is preceded by a minus sign (-), remove the
minus sign and the grouping symbol, but CHANGE the sign of all
terms inside that grouping symbol.
Examples:
1) (3 x  2)  ( x  7)
 3x  2  x  7
 4x  5
 final answer
2) (8a  6b  4c)  (5a  3b  2c)
 8a  6b  4c  5a  3b  2c
 13a  9b  6c  final answer
3) 4 x  x  (6 x  1)
 4 x  x  6 x  1
 4 x  7 x  1
 4x  7x 1
  3x  1
 final answer
4) 2a  34a  2a  3
 2a  34a  2a  3
 2a  32a  3
 2a  6a  9
 4a  9
 final answer
28
 Multiplication of polynomials
1.
In multiplying monomial by another monomial, multiply their numerical coefficients, and
multiply the literal coefficients using the multiplication law of exponents.
Examples: Find the product.
1) (5 x) (2 x)  10 x 2
2) (3a 5 ) (4a 6 )   12a 11
3) (10 x 3 y 2 z ) ( x 4 y )   10 x 7 y 3 z
4) (6a ) (3b)   18ab
5) (7 ax) (9bx 2 )   63abx 3
2.
In multiplying polynomial by a monomial, multiply the monomial to
each term of the polynomial.
Examples: Find the product.
1. (2 x)( x 2  3 x  5)
 2 x 3  6 x 2  10 x
2. (4a  7b) (3)
  12a  21b
3. (5 xy ) (2 x  3 y  1)
 10 x 2 y  15 xy 2  5 xy
4. (10a ) ( x  2 y  3 z )
 10ax  20ay  30az
3.
In multiplying polynomial by another polynomial, multiply each term
of the multiplicand to each term of the multiplier. Combine similar terms,
if possible.
29
Examples: Find the product.
1) ( x  5) ( x  6)
 x 2  6 x  5 x  30
 x 2  11x  30
 final answer
2) (2 x  3) (3 x  5)
 6 x 2  10 x  9 x  15
 6 x 2  x  15
 final answer
3) (a  b) (c  d )
 ac  ad  bc  bd
 final answer
4) ( x 2  2 x  3) (5 x  1)
 5x 3  x 2
10 x 2  2 x
15 x  3
5 x 3  9 x 2  13 x  3
 final answer
5) ( x 2  3 x  1) ( x 2  x  2)
 x 4  x3  2x 2
3x 3  3x 2  6 x
2x 2  6x  2
x 4  2 x 3  x 2  12 x  2  final answer
(2-2) Factorizing polynomials
Factorization of polynomials means to find the prime factors of the given polynomial. There are
various ways of factoring a polynomial and some are presented below:

Common Factor
ax  ay  az  a ( x  y  z)
30
Examples: Factor out the following.
1) 15 x  10 y  5  5 (3 x  2 y  1)
2) 2 x 3 y  8 xy 3  2 xy ( x 2  4 y 2 )
3) 4 x  10 y  2 (2 x  5 y )
4) 3 x 3 y  2 x  x (3 x 2 y  2)

Difference of Two Squares
x 2  y 2  ( x  y) ( x  y)
Examples: Factor out the following.
1) x 2  4
 x 2  22
 ( x  2) ( x  2)
 final answer
2) 100 x 4  1
 (10 x 2 ) 2  12
 (10 x 2  1) (10 x 2  1)
 final answer
3) 9  36a 2
 9 (1  4a 2 )
 9 (1  2a ) (1  2a )

 final answer
Trinomials
2
x  (a  b) x  ab  ( x  a) ( x  b)
31
Examples: Find the factors of the following.
1) x 2  7 x  12  ( x  4) ( x  3)
2) x 2  2 x  24  ( x  6) ( x  4)
3) x 2  10 x  25  ( x  5) ( x  5)
4) x 4  x 2  30  ( x 2  5) ( x 2  6)
5) x 2  x  20  ( x  5) ( x  4)
Exercises: Perform the indicated operation.
I – Find the sum.
1)  5 x
2)  2 y 3
3)  10 x 3 y
 4x
 y3
 4x3 y
4) 4a  b
a  3b
5)
3x 2  x  6
4x  3
II – Find the difference.
1)  3x
2)  2 y 3
3)  8 x 3 y
 4x
 7 y3
 5x 3 y
4) 4a  b
5)
3x 2  5 x  1
5a  6 b
x 2  4x  3
32
III – Simplify the following expressions by removing the grouping symbol/s.
1)
2)
3)
4)
5)
(3 x  5)  (4 x  2)
(a  6b  4c)  (5a  3b  c)
6 x  2 x  ( x  1)
3a  24a  a  5
a  2a  5  a  (3a  1)
IV – Find the product.
1) (3 x) (7 x)
2) (2a 4 ) (5a 7 )
3) (8 x 2 y 5 z ) ( x 3 y )
4) (3a ) (9b)
5) (2ax) (5bx 6 )
6.
7.
8.
9.
(4 x)( x 2  3 x  2)
(5a  2b) (6)
(3 xy ) ( x  5 y  2)
( 4a ) ( x  y  5 z )
10. (5 x 2 y ) (3 x  2 y  xy )
11. ( x  3) ( x  4)
12. ( x  5) (2 x  3)
13. (3x  1) (2 x  1)
14. ( x 2  2 x  5) (3x  1)
15. ( x 2  2 x  1) ( x 2  3x  2)
V – Factor out the following completely.
1. 18 x  36 y  6
2. 7 x 4 y  21xy 3
3. 8 x  30 y
4. 3 x 3 y  9 x
5. 25 x 3  15 x 2  10 x
33
6.
x 2  81
7. 64 x 4  9
8. 1  25a 4
9. 49 x 6  1
10. 100 x 4  81 y 2
11. x 2  12 x  35
12. x 2  2 x  63
13. x 2  13 x  40
14. x 2  8 x  7
15. x 4  5 x 2  36
VI – Practical applications
1.
In the WELDING workshop, a trainee worked x + 5 in Saturday, 3x – 2 in Sunday,
2x + 3 in Monday, and absent himself in Tuesday and Wednesday. How much
work did he do for that week?
2.
In Shinas VTC, the mechatronics workshop is rectangular whose length is x + 5 and whose
width is 2x – 1. What is the area of the mechatronics workshop?
(Note: Area of a rectangular = length times width).
3.
The total land area of the rectangular lot of Shinas VTC is 100x2 – 36y2 square units. If one
side is 10x + 6y, how long is the other side of the VTC?
In the ELECTRICAL workshop, the teacher told the trainees to finish individually the
4.
work 3x2 + 7x + 3. Nadir, one of the trainees, has finished x2 – 4x – 1 of the work.
How much is the unfinished work?
5.
In Shinas VTC, the teacher walked from faculty room to the workshop at 2x – 3y
distance, and went to the principal’s office at 5x + 7y. How far did the teacher
walked?
References:
Greer A. and Taylor G.W., BTEC First MATHEMATICS for Technicians, TJ International
Ltd, Padstow, Cornwall, 1982.
Vance, Elbridge P., Modern College Algebra, Addison-Wesley Publishing Company Inc.,
1983.
Systems Technology Institute Inc., College Algebra, STI, Inc. 1997.
Haag, Vincent H., Structure of Algebra, Addison-Wesley Publishing Company Inc., 1966.
34
Chapter 3
Solution of Equations and Inequalities
Solution of linear equations
solution of linear equations in one variable:
The equation 2x + 5 = 1 . is a linear equation in one variable x and is required:
Example: Find the value of x in the following equation:
2 x + 5 = 1→
2x=1–5 →2x=-4 →x= -2
Exercise : solve the following equations:
1)3+x=2
2)3x-1= 8
Solving Linear equations in two variable:
The equation 2x + y = 7 is a linear equation in two variable x and y are required:
Example: Solving Linear equations in two variables:
2x+y=7
Sol :
,
3x–y = 8
2 x + y = 7 (1)
3 x – y = 8 (2)
‫ـــــــــــــــــــــــــــــــــــــ‬
5x
= 15 →
by substitution 2x + y = 7 → 2 ( 3 ) + y = 7
S.S = {(3 , 1 )}
35
x=3
→6+y=7
→ y=1
Example: Solve the following equations
4 x + 5 y = 10
,
3 x + y = 13
Sol :
4 x + 5 y = 10 ( 1)
3 x + y = 13 (2)→ ×( – 5 )
‫ــــــــــــــــــــــــــــــــ‬
4 x + 5 y = 10
-15 x – 5 y = - 65
‫ـــــــــــــــــــــــــــــــــــــ‬
-11 x
= - 55
→x=5
by substitution 3 x + 5y = 13 → 3 ( 5 ) + 5 y = 13 → 15 + 5 y = 13
y = -2/5
5y=-2→
→ S . S = { ( 5 , -2/5
)}
Exercises: Solving Linear equations in two variables:
1) x–y=1
, x +y = 9
2)6x–y =5
, 4y -6 x = 1
3) x+3y=9
,
2x+y = 1
Solution of linear inequality in one variable
3 x + 5 > 11 is a linear inequality in one variable x and is required:
Example: Find the value of x in the following inequality:
3 x + 5 > 11
Sol: 3 x + 5 > 11 → 3 x > 11 – 5 → 3 x >6÷ 3) → x > 2
36
Example: Find the value of x in the following inequality:
1
x 1  2
2
×2 (  X–2≤ 4 X≤6
Sol:
Example: Find the value of x in the following inequality:
-2 > x - 5 > 1
Sol : - 2 + 5 > x > 1 + 5  3 > x > 6
Exercise: Find the value of x in the following inequality:
1)x+4<7
3)
2)2x–3>7
x- 2≤-3
4)
-1 > x - 2 > 1
Solving Quadratic Equation
1 ) by factorizing
2 ) by formula
1 ) by factorizing:
Ex: Solving the following Equations:
2
1) x – 5 x +6 = 0
2
2) x – 9 = 0
2
3 ) x + 6 x – 16
2
Sol : 1 ) x – 5 x +6 = 0
( x – 3 )=0
( x – 2 )=0
X–3=0
x- 2 = 0
X=3
X=2
→
SS={2,3}
2
Sol : 2) x – 9 = 0
( x + 3 ) =0
( x - 3 )=0
X=-3
Sol
X = 3 →
S. S = { 3 , - 3 }
2
: 3 ) x + 6 x – 16
( x + 8 ) =0
X=-8
( x – 2 )=0
x=2
37
→ S . S = { -8 , 2 }
2) By formula:
Equation can be solved by Quadratic Formula
ax+by+c=0
by using the formula
 b  b 2  4ac
x
2a
where:
a is X-Factor,
b is Y-Factor,
Ex : Solve the equation following by formula:
c the absolute term
2
x –2x- 2=0
Sol : a = 1
, b = -2
, C = -2
 b  b 2  4aC
x
2a
2  2 2  4  1  2
x
2 1
2  12
x
2
2  3 .5
x1 
 2.75
2
2  3 .5
x2 
 0.75
2
38
Exercise
1) Solving the following Equations by factorizing:
2
1 ) x + 7 x + 10 = 0
2
2 ) x – 5 x – 14 = 0
2
3)x –4=0
2) Solving the following Equations by formula :
2
1 ) x + 2x – 4 = 0
2
2)x + x -2 = 0
39
Chapter 4
Sequences and Series
I) Sequences:
1-Definition:
A sequence is a set of real numbers u1 , u 2 , u3 ,..........u n which is
arranged (ordered).
Example: 3,9,27,..........,3n
Each number u k is a term of the sequence.
We called u1 - First term and u 45 - Forty-fifth term
The n th term u n is called the general term(final term) of the sequence.
1 2 3 4 5 6 7
3 4 5 6 7 8 9
Example1: If , , , , , , is a sequence.
a. Find u1 , u 2 , u3 , u5 , u7
b. Find u n
2-Sequences types:
 Finite sequence
 Infinite sequence
Example2:
  2;  4;  6;  8;.............. infinite sequence .
1 1 1 1 1
finite sequence .
; ; ; ;
2 4 8 16 24
Example3: Write the 3 first terms of the sequences where the general term is:

a.
c.
b. un   2n 1
u n  3n  2
un  1 
1
n
n
2
d. u n   1
40
Example4: Write the 5 first terms of the sequences:
a. 3; 5; 7;.................
b. 5; 9;13;.................
c.  2; 4;  8;.................
Example5: Write the general term of the sequences:
a.
b.
c.
d.
1; 5; 9;13;17;.................
 1;1;  1;1;  1;.................
1; 8; 27;64;125;.................
0; 5;  10;  15;  20; .................
II) Series:
1-Definition:
The symbol ∑ (sigma) is called the summation sign.
This symbol will represents the sum of the first n terms as follows:
n
u1  u 2  u 3  .........  u n   u r
r 1
N.B: To express a series using summation sign ∑, we have to find the general term of
the sequence
2- Series types:

Finite series : it is same as finite sequence
 Infinite series: it is same as infinite sequence
Example6: Express the following using the summation sign ∑:
a. 1  2  3  4  ..............  10
b. 3  6  9  12  15  18  21
c.
1 1 1 1
    .........
2 4 8 16
Example7: Write all the terms of the series:
a.
b.
5
 2 1  r 
r 1
4
3
  1
n
n3
n 1
41
III) Arithmetic Sequences and series:
Consider the following lists of numbers
)i) 1,2,3,4,………
(ii) 50,40,30,20,……
(iii) -10,-8,-6,-4,…….
Each number in the above lists is called a term.
Note that
In (i) each term is 1 more than the term preceding it.
In (ii) each term is 10 less than the term preceding it.
In (iii) each term is obtained by adding 2 to the term preceding it.
1-Definition:
An Arithmetic sequence is a sequence in which the difference between any
two consecutive terms is a constant.
The constant difference is called the common difference and it is denoted by
the letter d ; ( d  u k 1  u k )
An arithmetic sequence with first term u1  a and common difference d , can
be written as follows: a; a  d ; a  2d ; a  3d ;.........; a  (n  1)d
The general term of an Arithmetic sequence is: u n  a  (n  1)d
Drill 1
For the arithmetic sequence 5,7,9,……
Write the first term a and the common difference d.
42
Drill 2
Which of the following list of numbers does form an A.S? If they form an A.S write
the next two terms.
(i) 4,9,14,19,……
(ii) -3,3,-3,3,-3,……
(iii) 1,1,1,2,2,2,3,3,3,…….
Drill 3
Write first four terms of the A.S when the first term a and the common difference d
are given as follows:(1) a=10 , d=10
(2) a=-4 , d=1
(3) a=1 , d=0.5
Drill 4
Which term of the A.S?
21,18,15,………is -81
Example8: Write the first 5 terms of the arithmetic sequence whose first term 3 and
common difference 4 .
Example9: Show that 2;1;4;7;............ is an arithmetic sequence and find its 10th
term.
Example10: Find the number of terms in the arithmetic series:
3  5  7  ...............  61
43
2- Sum of arithmetic series:
Theorem: The sum to n terms of an arithmetic sequence a, a  d , a  2d ,...........; p
n
2
Is given by S n  (a  p)
N.B:
where:
a =first term
;
p = final term (general term)
p  u n  a  (n  1)d , then the sum to n- terms is given by:
Sn 
n
(2a  (n  1)r )
2
Drill 5
1- Find the sum of the first 20 terms of the A.S 8,3,-2, …………….
2- Find the sum of the first 1000 positive integers.
3- Find the sum of first 20 terms of the list of numbers whose nth term is given by:
an= 3+2n
Example11: Find the sum of the following series:
a. 3;7;11;..........; n( 15)
b.
7
 3n  2
n 1
c. Odd natural numbers from 1 to 99 .
d. Even natural numbers from 2 to 100 .
44
IV) Geometric Sequences and Series:
Consider the following lists of numbers:
)i) 2,4,8,16,………
(ii) 3,12,48,192,……
(iii) -4,2,-1,1/2,…….
Each number in the above lists is called a term.
Note that
The ratio between any term and the preceding term is the same.
1-Definition:
A Geometric sequence is a sequence in which the ratio between any term and
its preceeding term is a constant.
The constant ratio is called the common ratio and it is denoted by the letter r
( r
Vk 1
)
Vk
A Geometric sequence with first term V1  a and common ratio r , can be
written as follows: a; ar; ar 2 ; ar 3 ;.............; ar n1
The general term of a Geometric sequence is: Vn  a r n1
Drill 1
For the G.S: 3,6,12,…….
Write the first term a and the common ratio r.
45
Drill 2
Which of the following list of numbers does form a G.S? If they form a G.S write the
next two terms.
(i) 100,50,25,…..
(ii) -3,3,-3,3,-3,……
(iii) 1,2,6,….
Drill 3
Write first four terms of the G.S when the first term a and the common ratio r are
given as follows:(1) a=10 ,r=2
(2) a=-4 ,r=0.5
(3) a=1 , r=-3
Drill 4
The first term of geometric sequence is 5 and its common ratio 2 write the first six
terms and what the order of the term whose value is 5120.
Example12: Find the common ratio and the 7th term of the geometric sequence
3,9,27,81,..........
Example13: Find the number of terms in the geometric sequence whose first term is
625 , the last term is 1 and common ratio is
1
.
5
Example14: Find the first term of a geometric sequence whose V5  311and V7  448
46
2- Sum of geometric series:
Theorem: The sum to n- terms of a geometric series whose first term is a and
common ratio is r is given by :
a(1  r n )
Sn 
1 r
; r 1
N.B: The sum of a geometric series whose first term is a and common ratio is r
and the last term is p is given by:
Sn 
a  pr
1 r
; r 1
Drill 5
1- Find the sum of the first 20 terms of the G.S 2,4,8,…….
2- Find the sum of first 20 terms of the list of numbers whose nth term is given by:
Example15: Find the sum of the first 6 terms of the geometric sequence whose first
1
2
term is 6 and common ratio is ( ) .
Example16: Find the first 7 terms of the geometric series:
 7  14  (28)  ..............
Example17: Find the sum of the geometric series:
1
1 1 1
1
   ............ 
2 4 8
64
Example18: Find the first term of a geometric series whose common ratio is (2) and
the sum of the first 6 terms is (63)
Example19: Find the first term in a geometric series whose sum is 240 ,the common
ratio is 3 and the last term is 162 .
47
Note that
1-To prove any sequence is an arithmetic sequence you must prove the difference of
any two consecutive terms is the same.
2- if a,b,c is an A.S then b=(a+c)/2 and b is called the arithmetic mean of a and c.
3  If a, b, c is a G.S then b  a.c
B is called geometric mean of a and b.
4- If the sequence has a last term then it is called finite sequence. and if it does not
have a last term it is called infinite sequence.
48
Chapter 5
Logarithms
Exponential and Logarithmic functions
I) Exponential function:
1-Definition: The exponential function is a real function. Its domain is IR and its range
is IR+ .
; a IR \ 1
f : IR  IR
f ( x)  a x
Example 1: Given the exponential function f with base 3 :
a. find f (1), f (3), f (2), f (0)
b. If f ( x) 
1
, find the value of x ?
81
N.B:
The symbol a n signifies to multiply a a number of n factors:
an = axaxax…xa


a =Base
n = Exponent
2- Law s of exponential:
If

a, b  IR ; n, p  z then:
1)
a n . a p  a n p
an
 a n p
p
a
5)
(a n ) p  a n. p 6)
a 0  1 2)
a
n
n
1
 n
a
an
a
   n 7)
b
b
3)
(a.b)  a . b 4)
n
n
n
a
49
n
p
 a n 8)
p

a  IR ; ‫ و‬p IN 
Example 2: Find the value
a-
32
1
5
b- 64
1
3
c- 3 2
Example3: Write in fractional exponents form
7
a-
3
42
b-
5
26
c-
43
Example4: Simplify the following expressions
A
1
3
 
x 5y
3
1
2 2
x  15 y
B
3
4

2
3
(81) 8
1
4 1  ( ) 3
3
5 2n2  7 4n
(49) 2 n  (25)12 n
C
3- Exponential equations:
 The exponential equation is any equation which contains variable in the
exponent for example x .
 Solve an exponential equation is to find the unknown variable x .
 We write the exponential equation in the form of a x  a c ,then:
a x  ac ; a  0  x  c
Example5: Solve the exponential equations
a- x 3   27
b- 64 x  4
c- 3 x  81
d- 132 x 1  5 2 x 1
e- 6 x  2  3 x  2
f- ( ) x 
N.B:
1
4
a x  b x ; a , b 0 ; a  b  x  0
Example6: Solve the equations
a- 1252 x3  59 x6
b- 4 x  17  2 x  16  0
Example7:
1
2
If f ( x)  (a x  a  x ) prove that f ( x  y) f ( x  y)  2 f ( x) f ( y)
Example8:
Prove that ( x  2 x  1)( x  x ) 1  1
1
x
50
1
256
4- Curves of exponential functions:
To draw the curve of any function,we take some numbers and find their images, we
obtain some ordered pairs representing points. We fix these points in the cartesian
coordinate and we connect them with a curved line.
Example9: Draw the curve of f ( x)  2 x .
x
3
2
1
0
1
2
1
4
1
2
1
2
4
f (x)
1
8
N.B:
 In the figure we draw the curve of the function g ( x)  2  x and we have
g ( x)  f ( x) then the curve of f and g are Corresponding with y -axis.
 All curves of exponential functions pass through the point (0;1) .
Example10: Draw the curve of the function f ( x)  3 x and deduce the curve of
f ( x)  3  x .
51
II) Logarithmic function:
1-Definition:
The Logarithmic function is a real function. Its domain is IR and its range is IR .
f  Log a : IR  IR
;
a IR \ 1
f ( x)  Log a ( x)
2-Laws of logarithmic functions :
a. Log a (xy )  Log a (x) + Log a ( y)
; x; y IR*
x
y
b. Log a ( )  Log a (x)  Log a ( y)
c. Log a ( x n )  nLog a x
1
x
e. Log a (a )  1
d. Log a ( )   Log a (x)
f. Log a (a n )  n
1
2
g. Log a ( x )  Log a ( x)
; Log a (n x ) 
1
Log a ( x)
n
N.B:
 Log e ( x)  Ln( x) ; ( e  2,71 ) (Normal function of logarithm)
 e Log( x )  x ;
Log (e y )  y

Loga 1  0

Log a a  1
Example11: Find the value
A  log 4 64  log 4 16  log 2 32
B
52
log 3 5  2 log 3 4
1
log 3 80
3
3-The logarithm and its relation with indices:
The logarithmic function is the inverse of the exponential function:
Log a ( y)  x  y  a x
Example11: Express in logarithmic form
a-
2 8
3
b- 5
3
1

125
1
4
c- 81  3
Example12: Express in exponential form
a-
log 3 81  4
b- log 2
1
 4
16
c- log 2 2 2 
3
2
4- Solution of Logarithmic equations :
Definition:
 The solution of logarithmic equation depends on the solution of exponential
equation
 To solve the logarithmic equation we express it in the exponential form and
then we find the desired variable .
Example13:
Find the solution set of the following equations:
a- Log 2 x  3
b- Log x 81 4
d- Log 5 ( x  1)  2
e- Log 2 ( x 2  1)  3
f- Log x1 27  3
h- Log3 ( x 1) 6
k- Log x (5x  6)  2
g- Log
x 2
125 2
53
c- Log3 x   2
5- Curves of logarithmic functions:
To draw the curve of any logarithmic function, we take some numbers and find their
images; we obtain some ordered pairs representing points. We fix these points in
the Cartesian coordinate and we connect them with a curved line then we obtain the
desired curve.
f ( x)  Log 2 x
Example14: Draw the curve of
x
f (x)
1
8
3
1
4
1
2
4
8
2
0
1
2
3
N.B:
 The curve of y  a x and the curve of y  Log a (x) are corresponding with the
axis
: yx
 All curves of logarithmic functions passes through the point (1;0) .
54
Exercises
Exponential and Logarithmic functions
Exercise1: Find the value
1
1
a) 81 4
b) 125 3
c) 5 3
Exercise2: Write in fractional exponents form
a)
6
b)
43
3
25
c)
5
42
Exercise3: Simplify the following expressions
A
1
5
 
x 5y
5
1
2 2
x  10 y
3
4

2
3
(256) 8
B
1
2 1  ( ) 3
4
5 3n  2  7 5 n
C
(49) n  (25) 2n
Exercise4: Find the simplest value of
1
4
a) 81  3
2
1
2
b) 49  27
2
3
5 n 3  9  5 n
c)
5 n 1  9 2
Exercise5: Solve the exponential equations
a) x 3   64
b) 64 x  2
c) 5 x  225
d) 5 2 x1  32 x1
e) 10 x2  3 x2
f) ( ) x 
1
3
1
81
Exercise6: Solve the exponential equations
a) 64 2 x3  49 x6
b) 9 x  36  3 x  243  0
c) 49 x  50  7 x  49  0
Exercise7: Draw the curve of f ( x)  4 x and deduce the curve of f ( x)  4  x
55
Exercise8: Express in logarithmic form
a) 3  27
b) 4
3
3
1
4
1

64
c) 16  2
Exercise9: Find the value
log 3 81  log 3 9  log 5 25
a)
b)
log 4 5  2  log 4 3
3
 log 4 45
2
3
 1 
log a    
2
 27 
Exercise10: If
find the value of a
Exercise11: Find the solution set of the following equations
a) Log 2 x 5
b) Log x 32 5
c) Log3 x   4
d) Log 2 (2 x  1)  6
e) Log 4 ( x 2  1)  2
f) Log x116  2
g) Log
x 5
1
2
12 2
h) Log 5 (3x  10) 0
k) Log x (3x  2)  2
Exercise12: Draw the curve of f ( x)  Log3 x and deduce the curve of y  3 x .
Exercise13:Choose the correct answer :
1)
Log5 5  .... a) 0
2)
Log31  ....
3) a) 4
b) 1
b) 1
c) 10
a) 1
c) 128
b) 3
d) 7
56
d) 25
c) 0
d) 4
Log2 (2)7  ....
Chapter 6
Permutation, Combinations
And Binomial Theorem
I)Introduction:
***Fundamental
Principle of Counting:
 If it is possible to achieve an operation with m method and to achieve a
second operation with n method and a third operation to them with p
method then these 3 operations can be achieved successively with m  n  p
methods .
 We can generalize the result for all successively operations
◙ Example1:
How many possible numbers between 100 and 1000 can be formed with
5 in the first decimal place?
● Application:
How many three digit numbers having three different digits can be
made from the set 9,3,8,6,5?
***Factorial
of a number:
If nIN  then factorial n is :
n ! n(n  1)(n  2)...........  3  2  1
 n(n  1) !
N.B:
0!  1
◙ Example2:
; 1! 1
Factorial 3 is 3! 3  2 1  6
Factorial 4 is 4! 4  3  2 1  24
Factorial 5 is 5! 5  4  3  2 1 120
● Application:
Find the value of:
10!
8!
; 6! 4! ;
6!3!
5!
57
II) Permutations:
Rule1: If E is finite set its elements’ number n then the number of its permutation
elements is n ! where n IN
◙ Example3:
What is the number of permutation methods of the set E  a, b, c, d 
Rule2: If E is finite set its elements’ number n then the number of its permutation
elements lemmas m elements each time is:
n!
;mn
(n  m)!
 n(n  1)(n  2)..........(n  1  m)
Pnm 
N.B:
Pn0  1 ;
Pn1  n ;
Pnn  n!
◙ Example4:
How many 3 digit numbers can be formed formed from the elements
E  1,2,3,4,5 without repetition of any number in each one.
● Application: 10
trainees participate in a race ,how many methods the trainees
can occur (get) the three first ranks .
◙ Example5:
What is the value of n if Pn2  30 ?
III) Combinations:
Rule3: If E is finite set its elements’ number n then the number of its permutation
elements lemmas m elements each time with out consider the order of elements is:
C
N.B:
m
n
Pnm
n!


m! m!(n  m)!
; mn
1
C n0  1 ; C n  n ; C nn  1
◙ Example6:
How many methods can we choose a subset formed by three numbers
from the set of numbers 1, 3,7,9.
◙ Example7:
Find the value of C 73 and C106
◙ Example8:
What is the value of n if Cn3 35 ?
58
IV) Binomial Expansion:
Rule4: In the expansion of the binomial a  bn where n IN there are (n  1) terms.
◙ Example9:
Expand a  b 2 and write the number of terms in it.
◙ Example10:
How many terms are there in the expansion of x  y 7 , 2a b4
Rule5: The expan of a  bn is :
n
a  b n   C nk a nk b k
k 0
 C n0 a n  C n1 a n 1b  C n2 a n  2 b 2  ..............  C nn 1 ab n 1  C nn b n
 an
◙ Example11:
 C n1 a n 1b  C n2 a n 2 b 2  ..............  C nn 1 ab n 1  b n
Expand: x  y 5
,
a  b4
N.B: The quantity t k 1  Cnk a nk b k is the general term of the expansion a  bn .
◙ Example12:
Find the 5 th term in the expansion of
a) a  b 6
b) x  y 10
● Application: What’s the value of :
a)
b)
n
C
k
n
C
k
n
k 0
n
k 0
(1) k
59
GEOMETRY
Chapter 7: Plane and Solid Geometry
Mahmood Badrawy (Saham V T C)
Chapter 8: Analytic Geometry
Rabie Soliman (Sur V T C)
Ridha Bechir Gharbi (Sur V T C)
60
Chapter 7
Plane and Solid Geometry
1- Area of Squares and Rectangles
61
62
2- Area of Triangles
63
64
65
3- Area of Parallelograms
66
67
4- Area of Trapezoids
68
69
5- Circumference and Area of Circles
70
71
6- Surface Areas of Prisms and Cylinders
72
73
74
7- Surface Area of Pyramids and Cones
75
76
77
78
8- Volume of Prisms and Cylinders
79
80
9- Volumes of Pyramids and Cones
81
82
10- Surface Area and Volume of Spheres
83
Example:
find area ?
Solution
Exercise : find the area of a square its perimeter =24 cm .
Example: find the area of the parallelogram in the opposite figure.
Area of parallelogram =
b×h=8×9=72 cm2
Example : - in the opposite figure
find the perimeter and area.
84
Example : find the area and circumference of the circle
Solution : The radius is half of the diameter.
Then r =5
Area =πr2=3.14×25=78.5
C.F=πd=3.14×10=31.4
Ex : in the opposite figure find the
lateral surface area, total surface area
and the volume.
Solution L.S.A=2πr×h=2×3.14×4×10=251.2 cm2
Area of two base =100.48cm2
T.S.A=351.68 cm2
Volume =502.4 cm3
Example: find the area and the volume of the sphere where its radius =7 cm
solution
area of a sphere =4πr2=4×
volume =
85
Example
The radius of opposite sphere is 4 feet.
Find the area of the of this sphere and the volume
Solution
Area of the sphere= 4r2=4×3.14×16=200.96 feet2
The volume =
feet3
=
Example: in the opposite figure find
The total area and the volume
3
3
Solutions
3cm
total surface area =6L2=6×3×3=54cm2
volume =L3=3×3×3=27cm3
The opposite figure shows a cuboid.
Given that the volume of the cuboid is 250
cm3, find the height of the cuboid.
Volume = length x breadth x height
Height=
5
86
5
Expression
Definitions
The edge
Intersection of two faces
The vertex
Intersection of three edges
Or Intersection of three faces
The solid
That object (thing )which occupies space
Cubic cm
Volume of a cube with edge length 1cm
Cubic m
Volume of a cube with edge length 1m
Volume
Number of cubic units in the solid
Circular sector
A portion of a circular region which is bounded
by an arc of the circle and two radii passing
through the end points of the arc.
The height of a triangle
The length of The line segment which drawn
from vertex perpendicular to its opposite side
The radius
Line segment joining the center and any point
on the circle
The chord
Line segment joining any two points on the
circle
The circle
Simple closed curve equidistant from a fixed
point
87
Perimeter is the word used to describe the distance around
the outside of a figure.
To find the perimeter, add together the lengths of all of the
sides of the figure.
Circumference is the word used to describe the distance
around the outside of a circle
Like perimeter, the circumference is the distance round the
outside of the figure. Unlike perimeter, in a circle there are
no straight segments to measure, so a special formula is
needed.
Use when you know the radius.
Use when you know the diameter.
88
Table of area
Area (triangle)
Area (rectangle)
or
Area (rectangle) = (length)•(width)
Area (square)
Area (parallelogram)
Area (trapezoid)
89
Area (rhombus)
d1=diagonal 1
d2= diagonal 2
or
Area (circle)
Remark to get the side length of a square
90
also
Table of Solids
Rectangular Solid
V=lwh
SA=2lh + 2hw + 2lw.
Cylinder
Sphere
Cone
..
Volume = 1/3 (area of base) (height)
SA=area of base +area of lateral
surface area
91
Exercises
1) Which biggest area of a square with side length 5cm or rectangle with dimensions 4
cm and 6 cm?
Area of the square =…….
Area of the rectangle = ……
The area of the ……… is bigger
2) Complete
Cube with edge length 20 cm. then
= ……. cm
2
b) The total surface area = …… cm
2
a) The face area
c) The volume
= ……
3) Complete
The perimeter of one face of a cube = 20 cm .then
2
= ….. cm
2
b) The total surface area = …… cm
2
c) The lateral surface area = …… cm
3
d) The volume
= ….. cm
a) The face area
92
4) Solve the following ?
1. Cube with edge length 1.2cm find its volume approximated to nearest cm 3 .
2. Cube with volume 1000 cm 3 find its edge length.
3. Which greater volume cube with edge length 7cm or cuboid with dimensions 4 ,
6 and 8 cm
4. If the sum of edges of a cube is 48cm find its volume
5. Find the volume of a cylinder with a radius r=1 m and height h=2 m. Find the
volume of a cone with a radius r=1 m and height h=1 m
5) Put true or false
 edge length
the height of a cuboid = volume  base area
1- the volume of a cube = face area
2-
3- cube is a cuboid with square base
(
)
(
)
(
)
4- the volume of a cuboid with square base of side length 5cm and height 7cm is
35 cm
3
(
)
5- the base area of a cuboid with rectangular base=length × width (
93
)
Chapter 8
Analytic Geometry
I) Cartesian coordinate:
The cartesian coordinate formed by two orthogonal axes ,the horizontal is ( X -axis)
and the vertical is ( Y -axis).
Y
M
( x, y)
B
A
1
X
0
1
C
D
(The orthogonal axes makes 4 quadrants in the plane:A;B;C;D)
Any point M in the plane is defined by an unique couple of real numbers ( x, y)
named coordinates of M .In contrary any couple ( x, y) is represented by an unique
point in the plane.
Example 1: Represent the points A(2,3) ; B(4,4) ; C(2,5) ; D(4,2) in the cartesian
coordinate.what do you observe?
N.B: The four quadrants A;B;C and D recognized by:




The quadrant A:
The quadrant B:
The quadrant C:
The quadrant D:
x 0 ; y 0
x  0 ; y 0
x 0 ; y  0
x 0 ; y0
94
II) Mid -point coordinate of a line segment:
If A( x1 , y1 ) and B( x2 , y 2 ) are the end points of the line segment A B and I ( x, y) is the
mid-point then:
x
x1  x 2
2
y
y1  y 2
2
Example 2: Find the coordinates of I mid-point of the line segment joining the points:
a. A(2, 3) ; B(4, 5)
b. A(1,4) ; B(1, 0)
c. A(0,  1) ; B(1, 2)
Drill
If m is the midpoint of pq where p(2,3) and q(5,1) find the coordinate of m.
III) Slope of a straight line:
Definition 1: The slope of a straight line L is the angle tangent that L makes with
the positive X -axis.
m  tan 
We write:
;
0  
y
y
L

L

x
x
95
Example 3: Find the slope of a straight line L which makes an angle of
a)
2
3
b) 60 
c) 130 
d)

4
with the positive direction of X -axis.
Definition 2: If A( x1 , y1 ) and B( x2 , y 2 ) are any two points on the straight line L which
is not vertical,its slope is given by:
m
y 2  y1
x 2  x1
Note that
* The slope of a line may be positive or negative or 0 or not defined.
For a horizontal line which is parallel to the x- axis the slope = 0 and vice versa .
* For a vertical line which is parallel to y- axis the slope is not defined and vice
versa.
* For the equation y = mx+b the slope =m.
Example 4: Find the slope of staight line t L hat passes through the two points:
a. A(1, 3) ; B(4,1)
b. A(2,3) ; B(4,  1)
Definition 3: If the equation of straight line L is ax  by  c  0 ; a  0 then the
slope of L is:
m
a
b
Example 5: Find the slope of the straight line L : 3x  2 y  6
96
IV) Straight line equation :
Theorem:
1) The equation of a straight line whose slope is m and passes through the
y  y1  m ( x  x1 )
point A( x1 , y1 ) is :
2) The equation of a straight line whose slope is m and its Y - intercept is b is
given by:
y  mx  b
3) The equation of a straight line whose X -intercept is a and Y intercept is b is
given by:
x
y

 1
a
b
Example 6: Find the equation of a straight line whose slope is (1) and passes
through the point (2,3) .
Example 7: Find the equation of a straight line whose slope is (3) and its intercept
with Y -axis is (4) .
Example 8: Find the equation of a straight line which passes through the points
(1,3) and (4,2) .
Example 9: Find the equation of a straight line whose X -intercept is 3 and Y
intercept is 4 .
Example 10: Find the equation of a straight line which passes through the point
(2,3) and makes angle of 45 with the positive X -axis.
V) Parallel and perpendicular lines:
Theorem:
If we have two lines L1 and L2 with slopes m1 , m2 respectively then:
1- L1 // L2 (L1 is parallel to L2) if and only if m1=m2.
L2 (  2- L1 L1 is perpendicular to L2) if and only if m1 × m2=-1
97
Drill
Find the slope of the lines which passes through the following points
1- (3,4),(2,5)
2- (1,6),(3,7)
3- (5,1),(5,-2)
4- (1,4),(3,4)
Drill
Prove that the points A(2,3),B(4,4) and C(8,6) are collinear .
Example 11: Find the value of a that makes L1 :4 x  a y  8 and L2 : 2 x  3 y  5  0
a. Perpendicular
b. Parallel
Example 12: By using the slope prove that the points A(1,3) ; B(3,7) ; C(7.5) are the
summits of a right triangle.
98
Exercises
Analytic Geomerty
Exercise1:
Represent the points A(4,3) ; B(2,5); C(4,1); D(3,2); E(0,2); F (3,0); G(4,0); H (0,4) in
the Cartesian coordinate .What do you observe?
Exercise2:
Find the coordinates of the mid-point of the line segment joining the points :
B(3, 5) ; A(1,  3) a)
B(2, 0) ; A(0,4) b)
B(2, 2) ; A(1,  1) c)
Exercise3:
Find the Slope of the straight line L making an angle of
45  d)
30  c)
120  b)

a) with the positive direction of X -axis.
3
Exercise4:
Find the slope of staight line L that passes through the two points:
B(4, 2) ; A(1, 3) a)
B(1,  1) ; A(2,0) b)
Exercise5:
Find the slope of the straight line L in the following cases.
 2 x  3 y  1 a)
 3 y  x  2  0 b)
99
Exercise6:
Find the equation of a straight line whose slope is (2) and passes through the
point (1,3) .
Exercise7:
Find the equation of a straight line whose slope is (1) and its intercept with Y -axis is
3.
Exercise8:
Find the equation of a straight line which passes through the points (1,2) and
(4,3) .
Exercise9:
Find the equation of a straight line whose X -intercept is (2) and Y intercept is 5 .
Exercise10:
Find the equation of a straight line which passes through the point (4,1) and makes
an angle of 30  with the positive X -axis.
Exercise11:
Find the value of k that makes L1 : 2 x  k y  4 and L2 : 3x  2 y  1  0
a) Perpendicular
b) Parallel
Exercise12:
Find the slope of straight line L1 knowing that L1// L2 and the equation of L2 is
3x  2 y  1  0
Exercise13:
a) Find the slope of the straight line L which passes through the points
A(1,3) and B(1,1) .
b) Find the equation of the straight line L.
c) Find the slope of the straight line L1 knowing that L1  L
100
d) Find the equation of the straight line L1 that passes through the point I the
mid- point of the line segment A B .
Exercise14:
Prove that the points A(2,4) ; B(2,3) ; C(5,3) represent the summits of a rightangled triangle.
Exercise15:
Prove that the straight line L1 passing through the two points (2,3),(4,5) is parallel to
the straight line L2 passing through the two points (1,5),(2,6)
Exercise16:
Prove that the straight line L1 whose equation is 2y-x=3 is parallel to the straight line
L2 whose equation is y=0.5x-5
Exercise17:
Prove that the straight line L1 passing through the two points (4,1),(7,-3) is
perpendicular to the straight line L2 passing through the two points (7,4),(3,1).
Exercise18:
Find the value of k if the two straight lines:
L1: y=2x-1 , L2 : y=kx+5 are :
1- parallel
2- perpendicular
Exercise19:
Write the equation of the line passing through the given point with the given slope (
write the final answer on the form y=mx+b)
1- p(3,2) ,m=2
2- p(2,0) ,m=-4/3
101
TRIGONOMETRY
Chapter 9: Trigonometric Functions
Mohammed Fatoh (Saham V T C)
Chapter 10: Solution of Triangle
Ridha Bechir Gharbi (Sur V T C)
Mohammed Fatoh (Saham V T C)
102
Chapter 9
Trigonometric Functions
The Pythagorean Theorem ( Pythagoras’ Theorem ) :
The ancient Egyptians had used a triangle made
by ropes with dimensions 3, 4, and 5 units of
length to construct a right angle to be used in
constructing vertical walls .That means that
the ancient Egyptians had known this
theorem before Pythagoras. Investigating the
idea :
103
Calculate the area of the smaller squares A and B and the larger square C for each
triangle above then complete the table:
Discuss figure 4 :
Remarks :
There is a special relationship between the lengths of the legs and the length of
the hypotenuse. This relationship is known today as the Pythagorean Theorem.
104
OR
In a right-angled triangle, the sum of the squares of the lengths of
the legs equals the square of the
length of the hypotenuse.
In other words ,
If a right-angled triangle has sides of lengths a, b and c, where c is
the length of the hypotenuse, then a 2  b 2  c 2 .
Example 1 :
How high up on the wall will a 20-foot ladder touch if the foot of the ladder is
placed 5 feet from the wall ?
Solution :
The ladder is the hypotenuse of a right triangle, so
a2  b 2  c 2
105
h
375  19.4 ft
The top of the ladder will touch the wall about 19.4 feet up from the ground.
Notice that the exact answer in this example is 375 However , this is a practical
application , so you need to calculate the approximate answer .
Example 2 :
Find the area of the rectangular rug if the width is 12 feet and the diagonal
measures 20 feet.
Solution :
Example 3 :
A 17 ft ladder leaning against a wall has its foot 8 ft from the base of the wall.
At what height is the top of the ladder touching the wall ?
106
Solution :
Let h be the height at which the ladder touches the wall.
We can assume that the ground makes a right angle with
the wall, as in the picture on the right. Then we see that
the ladder, ground, and wall form a right triangle with a
hypotenuse of length 17 meters ( the length of the
ladder )
and legs with lengths 8 meters and h meters. So by the
Pythagorean Theorem, we have :
h 2 + 82 = 172  h 2 = 289 - 64 = 225  h = 15 meters
107
Exercises :
In Exercises 1–11, find each missing length. All measurements are in
centimeters. Give approximate answers accurate to the nearest tenth of a
centimeter :
12) The diagonal of a square measures 32 meters .What is the area of the square?
13) What is the length of the diagonal of a square whose area is 64 cm2
108
14)
15)
109
16)
17)
A surveyor places poles at points A, B, and C in order to measure the distance
across a pond. The distances AC and BC are measured as shown. Find the distance
AB across the pond.
110
Degree measure ( Sexagesimal system ):
To measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal
parts; and we call each of those equal parts a "degree" . Its symbol is a small

:
1°
" 1 degree "
The full circle, then, will be 360° .
The measure of an angle has been expressed in degrees, minutes, and seconds.
/
One minute, denoted 1 , is such that
60/  1
, or 1/ 
//
One second, denoted 1 , is such that 60/ /  1/ , or 1/ / 
Then 19 degrees, 25 minutes, 13 seconds could be written as
This
D M
/
S //
1
 1 .
60
1
 1/ .
60
19 25/ 13/ / .
form was common before the widespread use of scientific calculators.
Now the preferred notation is to express fractional parts of degrees in decimal degree form.
The
D  M / S //
notation is still widely used in navigation .
Most calculators can convert
D M
/
S // notation to decimal degree notation and vice versa.
Example 1 :
Convert
5 42/ 30// to decimal degree notation.
Solution :
By using the calculator , the result is
5 42/ 30//  5.71
Example 2 :
Convert 64.18 to
D M
/
S //
notation .
Solution :
By using the calculator , the result is
64.18  64 10/ 48//
111
Remarks :
A degree
 

is defined as the measure of the central angle
subtended by an arc of a circle equal to
1
of the
360
circumference of the circle.
A minute
 
/
a minute, or
is
1
of a degree ; a second
60
 
//
is
1
of a degree.
3600
Exercise 1 :
Convert to decimal degree notation. Round to two decimal places :
Exercise 2 :
Convert to degrees, minutes, and seconds. Round to the nearest second :
1) 2.4
2) 67.84
3) 11.75
4) 20.14
5) 83.025
6) 47.8268
7) 29.8
8) 0.253
9) 0.9
10) 39.45
11) 30.2505
12) 17.6
112
1
of
60
Radian measure ( Circular system ):
Degree measure is a common unit of angle measure in many everyday applications. But in
many scientific fields and in mathematics (calculus, in particular), there is another
commonly used unit of measure called the radian.
To assign a radian measure to an angle  , consider to be a central angle of a circle of radius 1,
as shown in Figure 1 .
The radian measure of  is then defined to be the length of the arc of the sector. Because the
circumference of a circle is 2 r , the circumference of a unit circle ( of radius 1 ) is 2 .
This implies that the radian measure of an angle measuring 360 is 2 .
In other words, 360  2 radians or
113
Some Conversions Between Degrees And Radians :
Remarks :
1) An angle of

2 rad is the same as an angle of 360 , or an angle of
t rad  

rad is the same as a 180 angle. This suggests the formula
 rad 180

for converting radian measure t to degree measure  or vice versa.
2) You should know the conversions of the common angles shown in Figure 2 .
For other angles, use the fact that 180 is equal to  radians .
114
3) The radian–degree equivalents of the most commonly used angle measures are illustrated
in the following figures.
Example 1 :
Convert each of the following to radians :
Solution :
115
Example 2 :
Convert each of the following to degrees :
Solution :
Exercise 1 :
Exercise 2 :
116
Exercise 3 :
Exercise 4 :
Exercise 5 :
Express each of the following angles in radian measure :
Exercise 6 :
Express each of the following angles in degree measure :
Exercise 7 :
117
The Trigonometric Functions Of Acute Angles :
We begin our study of trigonometry by considering right triangles and acute angles measured in
degrees. An acute angle is an angle with measure greater than 0 and less than 90 . Greek letters
such as  ( alpha ),  ( beta ) and  ( theta ) are often used to denote an angle. Consider a right
triangle with one of its acute angles labeled  . The side opposite the right angle is called the
hypotenuse .
The other sides of the triangle are referenced by their position relative to the acute angle  .
One side is opposite  and one is adjacent to  .
The lengths of the sides of the triangle are used to define the six trigonometric ratios.
OR :
118
Example 1 :
In the right triangle shown at left, find the six trigonometric function
values
of : (a) 
and (b) 
Solution :
We use the definitions.
Remarks :
1) For any angle, the cosecant, secant, and cotangent values are the
reciprocals of the sine, cosine, and tangent function values, respectively.
2) If we know the values of the sine, cosine, and tangent functions of an angle, we
can use these reciprocal relationships to find the values of the cosecant, secant,
and cotangent functions of that angle .
119
Example 2 :
Given that sin  =
4
3
4
, cos  =
and tan  =
, find csc  , sec  and cot  .
5
5
3
Solution :
Example 3 :
If sin  =
6
7
and  is an acute angle, find the other five trigonometric function
values of  .
Hint
We know from the definition of the sine function that the ratio
6
7
is
opp
Using
hyp
this information, let’s consider a right triangle in which the hypotenuse has length
7 and the side opposite  has length 6. To find the length of the side adjacent to
 , we recall the Pythagorean theorem
120
Exercise 1 :
In Exercises 1–6, find the six trigonometric function values of the specified angle :
121
7. Given that sin  
8. Given that sin  
2
5
5
, cos   and tan  
, find csc  , sec  and cot  .
3
3
2
1
2 2
, cos  
and tan   2 2 , find csc  , sec  and cot  .
3
3
Given a function value of an acute angle, find the other five trigonometric
function values :
17. To find the length of a lake, a person set stakes at point A and made the following measurements:
a. What is the measure of angle BAC?
b. What is the length of the lake?
18.
122
19.
123
20.
21.
124
22.
23.
24.
A is an acute angle such that sin A 
2
. Find the values of the other trigonometric functions of A.
3
125
Trigonometric Functions Of Special Angles :
For certain special angles such as 30 , 45 and 60 which are frequently seen in
applications, we can use geometry to determine the function values.
A right triangle with a 45 angle actually has two 45 angles. Thus the triangle is
isosceles, and the legs are the same length. Let’s consider such a triangle whose
legs have length 1 . Then we can find the length of its hypotenuse, c , using the
Pythagorean theorem as follows :
Such a triangle is shown below. From this diagram, we can easily determine the
trigonometric function values of 45 :
It is sufficient to find only the function values of the sine, cosine, and tangent,
since the others are their reciprocals.
It is also possible to determine the function values of 30 and 60 :
Since we will often use the function values of 30 , 45 and 60 ,either the triangles
that yield them or the values themselves should be memorized.
126
Remarks :
1)
2)
127
3)
Example :
Find the value of each of the following without using the calculator :




1) sin 30 cos 60  sin 90  tan 45
cos 2 60  cos 2 30
2)
sec30 tan 30
Solution :




1) sin 30 cos 60  sin 90  tan 45 

1
1

 1
2
2
2
2
1 4  2 2 5  2 2

4
4
cos 2 60  cos 2 30

2)
sec 30 tan 30
1
 
2
Exercise 1 :
Prove each of the following :
1)
cos 60  2cos2 30 1
2)
sin 90  2sin 45 cos 45
3)
tan 60 
4)
cos90  cos2 45  sin 45
2
 3
1 3
 


 2   4 4  1 3
2
1
2
2
2

3
3
3
3
2
2 tan 30
1  tan 2 30
128
Exercise 2 :
Find the missing measures .Write all radicals in simplest form :
129
130
Exercise 3 :
Find the distance a across the river .
Exercise 4 :
The length of the shorter leg of a 30° - 60° - 90° triangle is 24 meters. Find the length
of the hypotenuse .
Exercise 5 :
At the same time that the sun's rays make a 60 angle with
the ground ,the shadow cast by a flagpole is 24 feet . To the
nearest foot , find the height of the flagpole .
131
Chapter 10
Solution of Triangle
Sine rule
In any triangle ABC ,
a
b
c


sin A sin B sin C
* Thus in any triangle , the sides are proportional to the sines of the opposite angles.
* The sine rule is used to solve triangle if
a) Two angles and one side is given .
b)Two sides and non-included angle is given
Example 1 :
In  EFG , e  4.56 , E  43 and G  57 . Find f and g .
Solution :
132
Using the law of sines :
Example 2 :
Given the triangle below, solve for the missing parts.
Solution :
133
Exercise 1 :
Given the triangle below, solve for the missing parts.
Exercise 2 :
abc is a triangle where
a  10 cm , A  30 , B  45 Find c , b
Exercise 3 :
In triangle ABC , a= 5.75 , B=650 and C = 420 . Find b and c .
134
.
Exercise 4 :
In triangle ABC , a = 15 , c= 17 and C = 1150 . Find angle A .
Remarks :
The surface area K of any  ABC is one half the product of the lengths of
two sides and the sine of the included angle.
K 
1
1
1
b c sin A = a c sin B 
a b sin C
2
2
2
Exercise 5 :
DEF is a triangle where
d  30 cm ,
D  60 , E  45
Find e , f and the
surface area of the triangle DEF .
Exercise 6 :
Prove that the area of a parallelogram is the product of two adjacent sides and the
sine of the included angle.
135
Exercise 7 :
136
Cosine rule
In any triangle ABC ,
a 2  b 2  c 2  2 b c cos A
b 2  a 2  c 2  2 a c cos B
c 2  a 2  b 2  2 a b cosC
And
cos A =
cos B =
cos C =
b 
2
 c    a 
2b c
2
2
a 
2
 c   b 
2ac
2
a 
2
 b   c 
2ab
2
2
2
* Thus, in any triangle , the square of a side is the sum of the squares of the
other two sides , minus twice the product of those sides and the cosine of
the included angle .
* When the included angle is 90 , the law of cosines reduces to the
Pythagoras ,s theorem .
* The cosine rule is used to solve triangle if:
a)Two sides and included angle given
b)Three sides are given
137
Example 1 :
ABC is a triangle in which a  32 , c = 48 , and
B = 125.2 Find b and A .
Solution :
Using the cosine rule :
b 
cos A =
2
 c   a   71   48   32 
=
 0.9273768
2b c
2  71 48
2
2
2
2
2
A  22
Example 2 :
RST is a triangle in which r = 3.5 , s = 4.7, and t = 2.8 . Find S and R
Solution :
138
.
Example 3 :
In triangle ABC given that a = 4, b = 6 and C=600 . Find c .
Example 4 :
In triangle ABC given that a = 7, b = 4 and c=5 . Find A
.
Example 5 :
In triangle ABC where a = 3, b= 4 and C = 650 . find c
.
Example 6 :
In triangle ABC where a = 40, c = 25 and B = 400 . Find b
.
Example 7 :
In triangle ABC given that a = 20, b = 15 and c = 12 . Find C .
Exercise 1 :
ABC is a triangle in which b  30 cm , c = 14 cm , and A = 60
Find a and B .
Exercise 2 :
DEF is a triangle in which d  17 cm , e  14 cm , f  15 cm
Find D and E .
139
Solving Right Triangles :
Since we can find function values for any acute angle, it is possible to solve right
triangles. To solve a triangle means to find the lengths of all sides and the
measures of all angles.
Example 1 :
In  ABC ( shown at right ) , find a, b, and B,
where a and b represent lengths of sides and B
represents the measure of B .
Here we use standard lettering for naming the sides
and angles of a right triangle:
Side a is opposite angle A, side b is opposite angle B,
where a and b are the legs, and side c, the hypotenuse, is opposite angle C, the right
angle.
Solution :
B = 180   90  61.7   28.3
140
Example 2 :
In  DEF ( shown at right ) ,find D and F .Then find d.
Solution :
F = 180   90  55.58   34.42
141
Exercises :
Exercise 7 :
Solve
 ABC
if
B  90 , C  50
Exercise 8 :
Exercise 9 :
142
and b  8 cm .
Solving Oblique Triangles :
To solve a triangle means to find the lengths of all its sides and
the measures of all its angles.
The trigonometric functions can also be used to solve triangles that are
not right triangles. Such triangles are called oblique . Any right triangle ,
or oblique, can be solved if at least one side and any other two measures are
known. The five possible situations are illustrated as follows:
In order to solve oblique triangles, we need to derive the law of sines and the law of
cosines. The law of sines applies to the first three situations listed above. The law of
143
cosines applies to the last two situations.The law of sines is used to solve
triangles given a side and two angles or given two sides and an angle opposite
one of them . The law of cosines is needed to solve triangles given two sides
and the included angle or given three sides.
Example 1 :
Solution :
Using the sine rule :
Thus ,
144
Example 2 :
Solution :
Using the cosine rule :
 r   t    s 
2
cos S =
2
2r t
 s   t    r 
2
cos R =
2
2
2
2s t
Example 3 :
ABC is a triangle where
the unknown angles
a  20 , b  25 , c  30 solve the triangle ABC by finding
145
Solution :
cos A =
b2  c2  a2
2bc
25 2  30 2  20 2 1125

 0.75
2  25  30
1500
A  41.4 
cos C =
20 2  25 2  30 2
125
a2  b2  c2
=

 0.125
2  20  25
1000
2ab
C  82.8
B  180   ( A  C ) = 180  ( 41.4   82.8 )  55.8
Example 4 :
ABC is a triangle where a = 30, b = 40 and C = 750 .Find the unknown side and the
angles
Solution :
c 2  a 2  b 2  2ab cos C  30 2  40 2  2  30  40  cos 75
c  45
cos A 
b2  c2  a2
40 2  45 2  30 2 2725

 0.7569
=
2bc
2  40  45
3600
A  40 .8
B  180  ( A  C )  180  ( 40 .8  75 )  64 .2
Example 5 :
Solve the following triangle A B C in which:
a = 30
B=750
C = 250
146
Solution :
A  180  (75  25 )  80
a
b
c
30
b
c





sin  sin  sin C
sin 80 sin 45 sin 25
30
b
30  0.7071

b
 b  21.54
sin 80 sin 45
0.9848
b
c
21.54  0.4226

c
 c  12.87
sin 45 sin 25
0.7071
Exercise 1 :
Exercise 2 :
Exercise 3 :
Three gears are arranged as shown in the figure
at right .Find the angle  .
147
Exercise 4 :
During a rescue mission, a Marine fighter pilot receives data on an unidentified
aircraft from an AWACS plane and is instructed to intercept the aircraft. The diagram
shown below appears on the screen, but before the distance to the point of
interception appears on the screen, communications are jammed. Fortunately, the
pilot re members the law of sines . How far must the pilot fly?
Exercise 5 :
148
Exercise 6:
In  ABC , three measures are given. Determine which law to use when solving the
triangle then solve  ABC :
Exercise 7 :
Exercise 8 :
149
Exercise 9 :
Exercise 10 :
Exercise 11 :
Exercise 12 :
150
Applications Of Solving Triangles :
Angles of Depression and Elevation :
Many applications with right triangles involve an angle of elevation or an angle of
depression . The angle between the horizontal and a line of sight above the
horizontal is called an angle of elevation .
151
The angle between the horizontal and a line of sight below the horizontal is
called an angle of depression. For example, suppose that you are looking straight
ahead and then you move your eyes up to look at an approaching airplane .
The angle that your eyes pass through is an angle of elevation. If the pilot of the
plane is looking forward and then looks down, the pilot’s eyes pass through an angle
of depression .
Remarks :
1) Since the vertical and horizontal directions are perpendicular, the elements of
problems dealing
with the relationship between lines of sight and the horizontal lead naturally to right
triangles.
2) Since both angles are measured from horizontal lines, which are parallel, the line of
sight AB isa transversal, and since alternate interior angles for parallel lines are equal,
 
152
Notice:
Suppose that we want to find the height of this tree.We mark point A and measure
how far it is from the base of the tree.Then we measure the angle of elevation from A
to the top of the tree.
h
 tan( )  h  x tan( )
x
we have measured x and  , so we can calculate tan( ) and thus we can find h , which
is the height of the tree.
Example 1 :
from apoint on the ground 12m away from the foot of tree, the angle of the
elevation of the top of the tree is 300 find its height .
C
B
A
Solution :
by using trigonometric ratios we write
BC
1
 tan 30  

AB
3
BC 
AB
3

12
3
 4 3  6.928  7 m
153
Example 2 :
From the top of a vertical cliff 40 m high, the angle of depression of an object that is
level with the base of the cliff is 34º. How far is the object from the base of the cliff?
Solution :
Let x m be the distance of the object
from the base of the cliff
* angle of depression= 34 
but APˆ O  BOˆ P
(Alternative angles)
then APˆ O  34 
* From the triangle APO ,we have:
tan 34  
40
40
40
x 


x
0.6745
tan 34
 x  59.30 m
So, the object is 59.3 m from the base of the cliff
Example 3 :
A tower stands vertically on the ground. From apoint on the ground, 20m away form
the foot of the tower ,the angle of elevation of the top of the tower is 66  what is the
height of the tower?
Example 4 :
John wants to measure the height of a tree. He walks exactly 100 feet from the base of
the tree and looks up. The angle from the ground to the top of the tree is 33 . How tall
is the tree?
154
Example 5 :
An airplane is flying at a height of 2 miles above the ground. The distance along the
ground from the airplane to the airport is 5 miles. What is the angle of depression
from the airplane to the airport?
Example 6 :
An aerial photographer who photographs farm properties for a real estate company
has determined from experience that the best photo is taken at a height of
approximately 475 ft and a distance of 850 ft from the farmhouse . What is the
angle of depression from the plane to the house ?
Solution :
The angle of depression from the plane to the
house,  , is equal to the angle of elevation
from the house to the plane, so we can use
the right triangle shown in the figure. Since
we know the side opposite B and the
hypotenuse, we can find  by using the sine
function .We first find sin 
sin   sin B 
475
 0.5588
850
Thus the angle of depression is approximately 34 .
155
Exercise 1 :
A logger walks off 40 ft from the base
of a tree and estimates the angle of

elevation to the tree , s peek to be 70 .
Approximately, how tall is the tree ?
Exercise 2
Exercise 2 :
What is the angle of elevation of
the sun when a 35- ft casts a 20-ft
shadow ?
Exercise 3 :
A person stands at the window of a building so that his eyes are 12.6 m above
the level ground in the vicinity of the building. An object is 58.5 m away from
the building on a line directly beneath the person. Compute the angle of
depression of the person’s line of sight to the object on the ground.
156
Exercise 4 :
Exercise 5 :
A man drives 500 m along a road which is inclined 20 to the horizontal. How high
above his starting point is he ?
Exercise 6 :
A tree broken over by the wind forms a right triangle with the ground. If the broken
part makes an angle of 50 with the ground and the top of the tree is now 20 m from
its base, how tall was the tree?
Exercise 7 :
Exercise 8 :
To measure cloud height at night, a vertical
beam of light is directed on a spot on the
cloud. From a point 135 ft away from the
light source, the angle of elevation to the
spot is found to be 67.35 . Find the height of
the cloud.
157
Exercise 9 :
Exercise 10 :
A flagpole casts a shadow 25 meters long when
the angle of elevation of the Sun is 40°.
How tall is the flagpole to the nearest meter?
Exercise 11 :
A surveyor is finding the width of
a river for a proposed bridge.
A theodolite is used by the surveyor to
measure angles. The distance from the
surveyor to the proposed bridge site is
40 feet. The surveyor measures a 50°
angle to the bridge site across the river.
Find the length of the bridge to the nearest foot.
158
Exercise 12 :
The angle of elevation from a small boat to the top
of a lighthouse is 25°. If the top of the lighthouse is
150 feet above sea level, find the distance from the
boat to the base of the lighthouse.
Exercise 13 :
A painter props a 20-foot ladder against a house. The angle it forms with the ground
is 65°. To the nearest foot, how far up the side of the house does the ladder reach?
Exercise 14 :
A surveyor is 85 meters from the base of a building.The
angle of elevation to the top of the building is 20°.
If her eye level is 1.6 meters above the ground, find the
height of the building to the nearest meter.
Exercise 15 :
A fire is sighted from a fire tower at an angle of
depression of 2°. If the fire tower has a height of 125
feet, how far is the fire from the base of the tower
round to the nearest foot?
Exercise 16 :
159
Exercise 17 :
Exercise 18 :
160
Exercises
Trigonometry
Exercise1:
a  10
In triangle ABC :
A  130 
B  20
find b with out using tables.
Exercise2:
Solve the triangle ABC by finding the unknown sides and angles
If a  5
b  12
A  60
Exercise3:
In triangle ABC
if a  6.25
B  73
C  45
find b and c .
Exercise4:
In triangle ABC
if
a  13
c  15
C  110
find angle A .
Exercise5:
In triangle ABC given that a  6
b8
C  80
find c .
Exercise6:
In triangle ABC given that a  7
b3
c5
find A .
Exercise7:
In triangle ABC where a  35
c  22
161
B  50
Find b
.
Exercise8:
ABC is a triangle where
a  18 , b  23 , c  35 solve the triangle ABC by finding
the unknown angles
Exercise9:
ABC is a triangle where
a  20 , b  25 , C  65 find the unknown side and the
angles.
Exercise10:
Solve the following triangle ABC if: a  20 , B  85 , C  15
Exercise11:
from apoint on the ground 10m away from the foot of tree, the angle of the
elevation of the top of the tree is 40  find its height .
C
10m
A
B
162
Exercise12:
From the top of a vertical cliff 50m high, the angle of depression of an object that is
level with the base of the cliff is 48 . How far is the object from the base of the cliff ?
Exercise13:
An airplane is flying at a height of 4 miles above the ground. The distance along the
ground from the airplane to the airport is 7 miles. What is the angle of depression
from the airplane to the airport?
a) 62.51
b) 22.33
c) 29.7 
d ) 21.8
e) 0.47 
Exercise14:
John wants to measure the height of a tree. He walks exactly 100 feet from the base
of the tree and looks up. The angle from the ground to the top of the tree is 33 .
How tall is the tree?
Exercise15:
A building is 50 feet high. At a distance away from the building, an observer notices
that the angle of elevation to the top of the building is 41 . How far is the observer
from the base of the building?
163
APPENDIX
DICTIONARY
Prepared by
Rabie Soliman
Senior Teacher of Math (Sur V T C)
Revised by Ghareeb Zaki
Curriculum Specialist of Math & Physics
164
Table of contents
Subject
Pages
List of symbols
166
Addition and subtraction
167
Algebra
167
Algebraic expressions
174
ANGLES
166
Arcs
178
Axes
178
Calculus
179
Circle
181
Complex numbers
182
Division
183
Ellipse
184
Energy
185
Equations
186
Factorization
187
Force
188
Fractions
190
Functions
191
Geometry
192
Groups
198
Interest
198
Lines
199
Matrices
201
Names
201
Numbers
202
Ratio
203
Sequences
204
165
Sets
205
Solid geometry
206
Statistics
207
Trigonometry
210
Variables
211
Vectors
212
List of symbols
Symbol

Is read as
Therefore ,so or then

Is an element of or belongs to

Is not an element of or does not belongs to

Empty set

Is a subset of

Is not a subset of

Union

Intersection
Parallel or is parallel to


Perpendicular or is Perpendicular to
Triangle
Right angle (of measure 90 )
Parallelogram
166
Addition and subtraction
‫ػالِخ اٌطشػ‬
Sign of subtraction
Subtraction
‫ؽشػ‬
Subtrahend
‫ػ‬ٚ‫اٌّطش‬
ِٕٗ ‫ػ‬ٚ‫اٌّطش‬
Minuend
) ‫شطت‬٠ – ‫ؾزف‬٠ ( ‫خزضي‬٠
Cancel
) ‫اخزضاي ( ؽزف – شطت‬
Cancellation
‫ اٌطشػ‬ٟ‫ثبل‬
Difference (remainder =left)
‫خ اٌطشػ‬١ٍّ‫ ػ‬ٟ‫االعزؼبسح(االعزالف)ف‬
Bridging in subtraction (borrow )
)‫اٌؾزف ( االعزجؼبد‬
Elimination
‫ اٌطشػ‬ٚ‫اٌؾزف ثبٌغّغ أ‬
Elimination by addition or subtraction
Algebraic operations
‫خ‬٠‫بد اٌغجش‬١ٍّ‫اٌؼ‬
Algebraic sum
ٞ‫ع اٌغجش‬ّٛ‫اٌّغ‬
‫غّغ‬٠
Add
‫خ اٌغّغ‬١ٍّ‫ػ‬
Addition
Additive identity
ٟ‫ذ اٌغّؼ‬٠‫اٌؼٕظش اٌّؾب‬
Additive inverse
ٟ‫ط اٌغّؼ‬ٛ‫اٌّؼى‬
‫اٌؾغبة‬
Arithmetic
Algebra
)ٗ‫س ئشبسر‬ٟٞ‫ٔمً(ٔمً ؽذ ِٓ ؽشف ألخش ثزغ‬
Transpose
ً٠ٛ‫رؾ‬
Transformation
ٟ‫ً ٕ٘ذع‬٠ٛ‫رؾ‬
Geometrical transformation
Trivial solution
)ٗ‫ؾ ( ربف‬١‫ؽً ثغ‬
Magic squares
‫خ‬٠‫اٌّشثؼبد اٌغؾش‬
‫ِمذاس‬
Magnitude
‫خ‬٠ٚ‫ذ‬٠ ‫ػٍّخ‬
Manual operation
ٟ‫غ‬١‫اٌٍّف اٌشئ‬
Master file
‫فزشح‬
Interval
167
‫ؽخ‬ٛ‫فزشح ِفز‬
Open interval
‫فزشح ِغٍمخ‬
Closed interval
‫ؽخ‬ٛ‫ ٔظف ِفز‬ٚ‫فزشح ٔظف ِغٍمخ أ‬
half open (closed) interval
ٜٛ‫ِغز‬
Level
ُ‫ز‬٠‫غبس‬ٌٍٛ‫ا‬
Logarithm
Table
‫ي‬ٚ‫عذ‬
Imaginary
ٍٟ١‫رخ‬
Index
‫شط‬ٙ‫ف‬
) x + y < 5 ( ‫ٕخ‬٠‫ِزجب‬
Inequation
‫ٕخ اٌّضٍش‬٠‫ِزجب‬
Triangle inequality
) x < y ( ‫خ‬٠ٚ‫ال ِزغب‬
Inequality

Plus infinity
-
Minus infinity
Infinity

Quantity
‫خ‬١ّ‫و‬
‫ٌخ‬ٛٙ‫خ ِغ‬١ّ‫و‬
Unknown quantity
‫خ‬ٙ‫ش ِزغ‬١‫خ غ‬١ّ‫و‬
Scalar quantity
ً‫شى‬
Figure
Kinds of the roots
‫س‬ٚ‫اع اٌغز‬ٛٔ‫أ‬
Solution set ( s s)
ً‫ػخ اٌؾ‬ّٛ‫ِغ‬
Using the formula
ْٛٔ‫ثبعزخذاَ اٌمب‬
Sum at roots
ٓ٠‫ع اٌغزس‬ّٛ‫ِغ‬
ٓ٠‫ؽبطً ػشة اٌغزس‬
Product at roots
‫خ‬١‫خ األسػ‬١‫اٌغبرث‬
Gravity
‫ذ‬٠‫اٌؼٕظش اٌّؾب‬
Identity element
‫ِٕطمخ ؽشعخ‬
Critical region
‫فه اٌشِض‬٠
Decode
‫وبف‬
Sufficient
ٟ‫ع اٌؾغبث‬ّٛ‫اٌّغ‬
Arithmetic sum
168
‫ً اٌغزس‬١ٌ‫د‬
Surd index
‫ػشة‬
Times
Midpoint
‫ٔمطخ إٌّزظف‬
Multiple
‫ِؼبػف‬
‫ِؼبػف ِشزشن‬
Common multiple
‫اٌّؼبػف اٌّشزشن األطغش‬
Lowest Common Multiple (L C M )
‫ؽبطً اٌؼشة‬
Product
Multiplication
‫خ اٌؼشة‬١ٍّ‫ػ‬
Multiplication Table
‫ي اٌؼشة‬ٚ‫عذ‬
ٟ‫ذ اٌؼشث‬٠‫اٌّؾب‬
Multiplicative identity
) ‫ط‬ٛ‫ش ( ِؼى‬١‫ٔظ‬
Inverse
ٟ‫ش اٌؼشث‬١‫إٌظ‬
Multiplicative inverse
‫اط‬ٚ‫اصد‬
Couple
‫اط‬ٚ‫رساع االصد‬
Arm of couple
Range
ٜ‫اٌّذ‬
Row
‫طف‬
Column
‫د‬ّٛ‫ػ‬
‫ػالِخ ايعزس‬
Root sign
Square root
ٟ‫ؼ‬١‫اٌغزس أٌزشث‬
Cube root
ٟ‫ج‬١‫اٌغزس اٌزىؼ‬
Resultant
‫ِؾظٍخ‬
ٟٕ‫شعُ ِٕؾ‬٠
Plotting a curve
) ‫ح ( أط‬ٛ‫ل‬
Power
‫ح‬ٛ‫اوجش ل‬
Greatest power
Precision
‫اٌذلخ‬
Prime
ٌٝٚ‫أ‬
ٌٟٚ‫ػبًِ أ‬
Prime factor
) ‫بد‬١‫بػ‬٠‫ اٌش‬ٟ‫ن ( ف‬ٛ‫ِفى‬
Expansion
‫خ االٔؼىبط‬١‫خبط‬
Reflexive relation
169
Symmetric relation
ً‫خ اٌزّبص‬١‫خبط‬
Transitive relation
ٞ‫خ اٌزؼذ‬١‫خبط‬
‫ة‬ٚ‫ِؼش‬
Factorial
‫ؽمً ِشرت‬
Ordered field
ٟ‫ؼ‬٠‫ص‬ٛ‫ر‬
Distributive
‫غ‬٠‫ص‬ٛ‫ْ اٌز‬ٛٔ‫لب‬
Distributive law
‫غ‬٠‫ص‬ٛ‫خ اٌز‬١‫خبط‬
Distributive property
‫أط‬
Exponent
ٝ‫ أع‬ٕٝ‫ِٕؾ‬
Exponential curve
‫ِغبٌطخ‬
Fallacy
ً‫ؽم‬
Field
‫اٌّغمؾ‬
Projection
First projection
‫ي‬ٚ‫ِغمؾ أ‬
Second projection
ْ‫ِغمؾ صب‬
Law of indices
‫ْ األعظ‬ٛٔ‫لب‬
Left hand side
‫غش‬٠‫ايؽشف األ‬
Right hand side
ّٓ٠‫اٌطشف األ‬
Reciprocal image
‫خ‬١‫سح ػىغ‬ٛ‫ط‬
‫اخزضاي‬
Reduction ( reducing )
‫ػاللخ رىبفإ‬
Relation of equivalence
‫ت‬١‫ػاللخ رشر‬
Relation of order
) ً‫ٕفظً ( لبثً ٌٍفظ‬٠
Separable
)‫شاد‬١‫فظً(ِضً فظً اٌّزغ‬
Separation
‫ٔمؾ ِٕفظٍخ‬
Separated points
‫غ‬٠ٛ‫رؼ‬
Substitution
‫غ‬١ّ‫ػالِخ اٌزغ‬
Summation sign
Double point
‫اط‬ٚ‫ٔمطخ اصد‬
Initial value
‫خ‬١‫ّخ اثزذائ‬١‫ل‬
‫ِٕبظش‬
Corresponding
170
Master data
‫خ‬١‫غ‬١‫بٔبد سئ‬١‫ث‬
Determinant
‫ِؾذد‬
‫ػشة اٌّؾذداد‬
Multiplication(product) of determinants
‫ِؾذد اٌّؼبِالد‬
Determinant of the Coefficients
ُ‫سل‬
Digit
)‫بس‬١‫ّخ اٌّطٍمخ ( ِؼ‬١‫اٌم‬
Absolute value ( modulus )
ُ‫ز‬٠‫غبس‬ٌٍٛ‫أعبط ا‬
Base of logarithm
Base of a power
‫أعبط األط‬
Binomial
ٓ٠‫راد اٌؾذ‬
ٓ٠‫ن راد اٌؾذ‬ٛ‫ِفى‬
Binomial expansion
False
‫خبؽئ‬
Foreword
‫ِمذِخ‬
Substitution set
‫غ‬٠ٛ‫ػخ اٌزؼ‬ّٛ‫ِغ‬
Vertical motion
‫خ‬١‫اٌؾشوخ اٌشأع‬
‫ؼ‬١‫اؽذ اٌظؾ‬ٌٛ‫س ا‬ٚ‫عز‬
Roots of unity
‫ي‬ٛٙ‫ِغ‬
Unknown
‫ؽذح ِشثؼخ‬ٚ
Square unit
‫خ‬١‫عشػخ اثزذائ‬
Initial velocity
‫خ‬١‫بئ‬ٙٔ ‫عشػخ‬
Terminal velocity
‫عطخ‬ٛ‫اٌغشػخ اٌّز‬
Mean speed
)‫عشػخ ِٕزظّخ (عشػخ صبثزخ‬
Uniform speed
‫ذ‬١‫ؽ‬ٚ
Unique
‫ذ‬١‫ؽ‬ٚ ً‫ؽ‬
Unique solution
‫خ‬١‫ٔمطخ صالص‬
Triple point
Trisection
)‫خ‬٠ٚ‫ أعضاء ِزغب‬3‫ء‬ٟ‫ُ اٌش‬١‫ش( رمغ‬١ٍ‫رض‬
Topology
)‫خ ٌإلشىبي‬١‫ٕذع‬ٌٙ‫ اٌخظبئض ا‬ٟ‫جؾش ف‬٠ ‫بد‬١‫بػ‬٠‫ فشع س‬ٛ٘ ( ‫ب‬١‫ع‬ٌٛٛ‫ث‬ٛ‫ر‬
‫ؾ‬١‫ش ثغ‬١‫ؽً غ‬
Non- trivial solution
Container
ْ‫خضا‬
Plot
ُ‫شع‬٠
171
Drawing scale
ُ‫بط اٌشع‬١‫ِم‬
Meaningless
ٕٝ‫ظ ٌٗ ِؼ‬١ٌ
‫اؽذ‬ٚ ً‫ شى‬ٟ‫اسعُ ف‬
Graph in one diagram
‫اٌغجش اٌّغشد‬
Abstract algebra
ٟ‫بػ‬٠‫لغ اٌش‬ٛ‫اٌز‬
Mathematical expectation
ٟ‫بػ‬٠‫االعزمشاء اٌش‬
Mathematical induction
ٟ‫بػ‬٠‫إٌّطك اٌش‬
Mathematical logic
ٟ‫بػ‬٠‫رط س‬ّٛٔ
Mathematical model
ٟ‫بػ‬٠‫ً اٌش‬١ٍ‫اٌزؾ‬
Mathematical analysis
Logic value
‫خ‬١‫ّخ ِٕطم‬١‫ل‬
Absolute constant
‫صبثذ ِطٍك‬
)10 ‫خ ( األعبط‬٠‫بد‬١‫زّبد اػز‬٠‫غبس‬ٌٛ
Common logarithms
‫ك‬١‫اف‬ٛ‫اٌز‬
Combinations
Permutation
ً٠‫رجذ‬
Permute
‫جذي‬٠
Common
‫ِشزشن‬
Element
‫ػٕظش‬
‫خ‬٠ٚ‫س ِزغب‬ٚ‫عز‬
Equal roots
Continued equality
‫اح ِغزّشح‬ٚ‫ِغب‬
(A=b=c=d)
‫اٌؾزف ثبٌّمبسٔخ‬
Elimination by comparison
‫غ‬٠ٛ‫اٌؾزف ثبٌزؼ‬
Elimination by substitution
‫اٌّشافك‬
Conjugate
‫لشاس‬
Decision
ٟ‫لشاس ِٕطم‬
Logical decision
‫ئصاؽخ‬
Displacement
ٓ١‫ٓ ٔمطز‬١‫اٌجؼذ ث‬
Distance between two points
Estimate
‫ش‬٠‫رمذ‬
Evaluate
‫ؾغت‬٠
‫س‬ٚ‫غبد اٌغز‬٠‫ئ‬
Evolution
172
Factorial
‫ة‬ٚ‫ِؼش‬
Reaction
ً‫سد اٌفؼ‬
ً‫فؼ‬
Action
‫خ‬١ٍّ‫ػ‬
Operation
‫خ‬١‫خ صٕبئ‬١ٍّ‫ػ‬
Binary operation
‫خ‬١‫بع‬١‫سح ل‬ٛ‫ط‬
Standard form
‫خ‬١‫بد األعبع‬١ٍّ‫اٌؼ‬
Fundamental operations
Irreducible radical
)
5 ً‫خ ِض‬٠‫ش عزس‬١‫سح غ‬ٛ‫ّىٓ وزبثزٗ ثظ‬٠ ‫ ال‬ٞ‫ اٌغزس اٌز‬ٛ٘( ‫ؾ‬١‫ش لبثً ٌٍزجغ‬١‫عزس غ‬
‫س‬ٚ‫اعزخشاط اٌغز‬
Extraction of roots
) ‫ّب ٘زا اٌغزس‬ٕٙ١‫ٕؾظش ث‬٠ ٓ١‫ٓ ِزمبسث‬٠‫غبد ػذد‬٠‫خ ئ‬١ٍّ‫ؼضي اٌغزس (ػ‬٠
Isolate a root
Abstract
‫ِغشد‬
Respect to
‫ؽجمب ٌـ‬
Participated
‫شبسن‬٠
‫خ‬١‫بد اٌؾغبث‬١ٍّ‫ت اٌؼ‬١‫رشر‬
Order of mathematical operations
ٟٔ‫ب‬١‫ً اٌج‬١‫اٌزّض‬
Graphical representation
‫بط‬١‫ِم‬
Scale
‫عؾ‬ٛ‫ُ ِز‬١‫ِغزم‬
Midline
‫ػذد األثؼبد‬
Dimensionality
Geometric solution
)ٟٔ‫ب‬١‫ ( ث‬ٟ‫ؽً ٕ٘ذع‬
Permissible
) ٓ‫ػ ثٗ ( ِّى‬ّٛ‫ِغ‬
‫آٌخ ؽبعجخ‬
Calculator
‫ؽغبة‬
Calculation
ٟ‫ىبسر‬٠‫ؽبطً اٌؼشة اٌذ‬
Cartesian product
‫ض‬١ٌّّ‫ا‬
Discriminant
ْ‫ِش‬
Elastic
‫ٔخ‬ٚ‫ِش‬
Elasticity
Distance – time curve
ِٓ‫اٌض‬ٚ ‫ اٌّغبفخ‬ٟٕ‫ِٕؾ‬
Natural logarithms
‫خ‬١‫ؼ‬١‫زّبد اٌطج‬٠‫غبس‬ٌٍٛ‫ا‬
173
‫ِخطؾ‬
Diagram
ُٙ‫ع‬
Arrow
Arrow diagram
ّٟٙ‫ِخطؾ ع‬
Center of a curve
ٟٕ‫ِشوض ِٕؾ‬
Corollary
‫غخ‬١‫ٔز‬
Principle
) ‫لبػذح ( لبثٍخ ٌإلصجبد‬
Converse
‫ػىظ‬
Distance
‫ِغبفخ‬
‫خ‬٠ٚ‫أعضاء ِزغب‬
Equal parts
Method
‫مخ‬٠‫ؽش‬
Relationship (Relation)
‫ػاللخ‬
Note that
ْ‫الؽع أ‬
Together
ً ‫ِؼب‬
Therefore
‫ٌزٌه‬
Determine
) ‫ٓ ( ؽذد‬١‫ّػ‬
‫ ٌٍٕمطخ‬ٟٕ١‫األؽذاس اٌغ‬
x-coordinate ( abscissa )
‫ ٌٍٕمطخ‬ٞ‫األؽذاس اٌظبد‬
y-coordinate
‫اٌؼىظ ثبٌؼىظ‬
Vise versa
‫خ‬١‫بر‬١‫خ ؽ‬١‫بػ‬٠‫ِغبئً س‬
Mathematical life problems
‫ط‬ٛ‫ِؾغ‬
Concrete
Imaginary roots
‫خ‬١ٍ١‫س رخ‬ٚ‫عز‬
Numerical measure
ٞ‫بط ػذد‬١‫ل‬
Algebraic expressions
) ٞ‫ؽذ ػجبسح (ؽذ ِمذاس عجش‬
Term of an expression
ٞ‫ؽذ عجش‬
Algebraic term
ٞ‫اٌّمذاس اٌغجش‬
Algebraic expression
Adding and
expressions
subtracting
‫خ‬٠‫ش اٌغجش‬٠‫ؽشػ اٌّمبد‬ٚ ‫عّغ‬
algebraic
174
‫خ‬٠‫د اٌغجش‬ٚ‫ػشة اٌؾذ‬
Multiplying algebraic terms
Multiplying an algebraic term by an algebraic expressions
‫ ِمذاس‬ٟ‫ ف‬ٞ‫ػشة ؽذ عجش‬
ٞ‫عجش‬
‫اٌؼشة ثّغشد إٌظش‬
Multiplying directly ( by inspection )
Sign
‫ئشبسح‬
Simplification
‫ؾ‬١‫رجغ‬
Solution
ً‫اٌؾ‬
Triple
ٟ‫صالص‬
Trinomial
‫د‬ٚ‫ اٌؾذ‬ٟ‫صالص‬
Absolute term
‫اٌؾذ اٌّطٍك‬
Algebraic symbols
‫خ‬٠‫ص اٌغجش‬ِٛ‫اٌش‬
Solve algebraically
‫ب‬٠‫ؽً عجش‬
‫ب‬١ٔ‫ب‬١‫ؽً ث‬
Solve graphically
‫غ‬٠‫ص‬ٛ‫اٌز‬
Distribution
Distribution
addition
Coefficient
of
multiplication
‫ اٌغّغ‬ٍٝ‫غ اٌؼشة ػ‬٠‫ص‬ٛ‫ر‬
over
ًِ‫ِؼب‬
) ‫ؾ‬١‫( اٌجغ‬
Least common denominator
‫اٌّمبَ اٌّشزشن األطغش‬
‫د‬ٚ‫شح اٌؾذ‬١‫دسعخ وض‬
Degree of polynomial
) ٗ‫ش ِزشبث‬١‫ِخزٍف ( غ‬
Dissimilar
‫خ‬ٙ‫د ِزشبث‬ٚ‫ؽذ‬
Similar ( like) terms
‫خ‬ٙ‫ش ِزشبث‬١‫د غ‬ٚ‫ؽذ‬
Dissimilar (unlike) terms
‫ط‬ٚ‫عزس ِضد‬
Double root
‫خ‬١ٔ‫ اٌذسعخ اٌضب‬ٕٝ‫ِٕؾ‬
Quadric ( quadratic) curve
ٟ‫ج‬١‫ اٌزىؼ‬ٕٝ‫إٌّؾ‬
Cubic curve
‫ي‬ٛٙ‫ِغ‬
Unknown
Value
‫ّخ‬١‫ل‬
Satisfy
‫ؾمك‬٠
Verify
‫ؽمك‬
‫د‬ٚ‫غ اٌؾذ‬١ّ‫رغ‬
Grouping terms
175
‫ِزغبٔظ‬
Homogeneous
Homogeneous algebraic Polynomial
‫د٘ب ِٓ ٔفظ اٌذسعخ ثبٌٕغجخ‬ٚ‫غ ؽذ‬١ّ‫ْ ع‬ٛ‫د رى‬ٚ‫شح ؽذ‬١‫ وض‬ٟ٘ ( ‫خ اٌّزغبٔغخ‬٠‫د اٌغجش‬ٚ‫شح اٌؾذ‬١‫وض‬
)‫رح ِؼب‬ٛ‫شاد ِأخ‬١‫غ اٌّزغ‬١ّ‫ٌغ‬
Involution
ٞٛ‫ اٌم‬ٌٝ‫اٌشفغ ئ‬
Irreducible
) ً١ٍ‫ؾ ( اٌزؾ‬١‫ش لبثً ٌٍزجغ‬١‫غ‬
‫ؾ‬١‫ش لبثٍخ ٌٍزجغ‬١‫د غ‬ٚ‫شح ؽذ‬١‫وض‬
Irreducible polynomial
Multiplicand
ٛ٘ 7 ‫ فبْ اٌؼذد‬4 × 7
‫ؼشة ثؼذد آخش فّضال ئرا ػشة اٌؼذد‬٠ ٞ‫ اٌؼذد اٌز‬ٛ٘ ‫ة‬ٚ‫ِؼش‬
‫ة‬ٚ‫اٌّؼش‬
Multiplicator
ٗ١‫ة ف‬ٚ‫اٌّؼش‬
Numerical analysis
ٞ‫ً اٌؼذد‬١ٍ‫اٌزؾ‬
Numerical coefficient
ٞ‫ِؼبًِ ػذد‬
Respectively
‫ت‬١‫ اٌزشر‬ٍٝ‫ػ‬
‫سح‬ٛ‫( ثّغؾ ) اثغؾ ط‬
Simplify
Rule of signs
‫لبػذح اإلشبساد‬
Closure property
‫خ االٔغالق‬١‫خبط‬
‫اٌذِظ‬
Association
‫ اٌذِظ‬ٞ
‫خبصح‬
Associative property
Commutation
‫اإلثذاي‬
Distribution
‫غ‬٠‫ص‬ٛ‫اٌز‬
)‫ش أعب‬١‫ٗ اٌّزغ‬١‫ْ ف‬ٛ‫ى‬٠ ‫ ؽذ‬ٛ٘ ( ٝ‫ؽذ أع‬
Exponential term
Angles
‫خ‬٠ٚ‫صا‬
Angle
Interior angles
‫خ‬١ٍ‫ب داخ‬٠‫ا‬ٚ‫ص‬
Exterior angle
‫خ خبسعخ‬٠ٚ‫صا‬
Obtuse angle
‫خ ِٕفشعخ‬٠ٚ‫صا‬
Right angle
‫خ لبئّخ‬٠ٚ‫صا‬
Acute angle
‫خ ؽبدح‬٠ٚ‫صا‬
Reflex angle
‫خ ِٕؼىغخ‬٠ٚ‫صا‬
‫ ٔظف دائشح‬ٟ‫ِخ ف‬ٛ‫خ ِشع‬٠ٚ‫صا‬
Angle in semicircle (inscribed angle )
176
‫ؽ‬ٚ‫خ ٌٍّخش‬١‫خ ٔظف اٌشأع‬٠ٚ‫اٌضا‬
Semi-vertical angle of a cone
‫ّخ‬١‫خ ِغزم‬٠ٚ‫صا‬
Straight angle
‫خ‬٠ٚ‫سأط اٌضا‬
Vertex of angle
‫خ سأط اٌّضٍش‬٠ٚ‫صا‬
Vertical angle of a triangle
Alternate angles
ْ‫زبْ ِزجبدٌزب‬٠ٚ‫صا‬
Adjacent angles
ْ‫سرب‬ٚ‫زبْ ِزغب‬٠ٚ‫صا‬
complementary angles
ْ‫زبْ ِززبِزب‬٠ٚ‫صا‬
Supplementary angles
ْ‫زبْ ِزىبٍِزب‬٠ٚ‫صا‬
Corresponding angles
ْ‫زبْ ِزٕبظشرب‬٠ٚ‫صا‬
‫اؽذح ِٓ اٌمبؽغ‬ٚ ‫خ‬ٙ‫ ع‬ٟ‫زبْ ف‬١ٍ‫زبْ داخ‬٠ٚ‫صا‬
Interior angles on the same side of
transversal
Vertically opposite angles
‫زبْ ِزمبثٍزبْ ثبٌشأط‬٠ٚ‫صا‬
Coterminal angles
) ‫ب ِزىبفئخ ( ِزبخّخ‬٠‫ا‬ٚ‫ص‬
)ٓ١ّ١‫ٓ ِغزم‬١‫خ ث‬٠ٚ‫خ( صا‬٠ٛ‫خ ِغز‬٠ٚ‫صا‬
Plane angle
Angle of friction
‫خ االؽزىبن‬٠ٚ‫صا‬
Arms of an angle
‫خ‬٠ٚ‫ػٍؼب اٌضا‬
‫خ‬٠ٚ‫ِٕظف اٌضا‬
Bisector of an angle
Central angle
‫خ‬٠‫خ اٌّشوض‬٠ٚ‫اٌضا‬
Angle at circumference
‫خ‬١‫ط‬١‫خ اٌّؾ‬٠ٚ‫اٌضا‬
ٞ‫ش اٌذائش‬٠‫خ ةاٌزمذ‬٠ٚ‫بط اٌضا‬١‫ؽذح ل‬ٚ
Radian
‫خ‬٠‫خ اٌظفش‬٠ٚ‫اٌضا‬
Zero Angle
ٓ١‫ٓ ِزىبفئز‬١‫ز‬٠ٚ‫صا‬
Equivalent angles
‫ب اٌخبطخ‬٠‫ا‬ٚ‫اٌض‬
Special angles
‫خ‬٠ٚ‫ ٌضا‬ٟ‫إٌّظف اٌخبسع‬
External bisector
‫خ االٔؾشاف‬٠ٚ‫صا‬
Angle of deflection
ٞ‫خ االخزالف اٌّشوض‬٠ٚ‫صا‬
Eccentric angle
Quadrantal angles
ٗ١‫ب سثؼ‬٠‫ا‬ٚ‫ص‬
Consecutive angles
‫خ‬١ٌ‫ب ِززب‬٠‫ا‬ٚ‫ص‬
‫خ ِغغّخ‬٠ٚ‫صا‬
Solid angle
177
Arcs
‫ط‬ٛ‫ل‬
Arc
‫ط دائشح‬ٛ‫ل‬
Arc of a circle
Major arc
‫ط األوجش‬ٛ‫اٌم‬
Minor arc
‫ط األطغش‬ٛ‫اٌم‬
‫ اٌذائشح‬ٟ‫ط األطغش ف‬ٛ‫اٌم‬
Minor arc of a circle
Axes
‫س‬ٚ‫اٌّؾب‬
Axes
ٟ‫س اٌشأع‬ٛ‫اٌّؾ‬
Vertical axis
ٟ‫س األفم‬ٛ‫اٌّؾ‬
Horizontal axis
Rectangular axes
‫س لبئّخ‬ٚ‫ِؾب‬
Axis of symmetry
ً‫س اٌزّبص‬ٛ‫ِؾ‬
‫س اٌمطغ اٌّىبفئ‬ٛ‫ِؾ‬
Axis of the parabola
x-axis
‫ٕبد‬١‫س اٌغ‬ٛ‫ِؾ‬
y-axis
‫س اٌظبداد‬ٛ‫ِؾ‬
First quadrant
‫ي‬ٚ‫اٌشثغ األ‬
Second quadrant
ٟٔ‫اٌشثغ اٌضب‬
Third quadrant
‫اٌشثغ اٌضبٌش‬
Fourth quadrant
‫اٌشثغ اٌشاثغ‬
‫ ع‬، ‫ ص‬، ‫س ط‬ٚ‫ِؾب‬
X-,Y-,Z-, Axes
X,Y,Z, co – ordinates
‫ ع‬، ‫ ص‬، ‫بد ط‬١‫ئؽذاص‬
Oblique coordinates
) ‫ب لبئّخ‬ِٕٙ ٓ٠‫س‬ٛ‫ٓ وً ِؾ‬١‫خ ث‬٠ٚ‫ْ اٌضا‬ٛ‫ ال رى‬ٟ‫س اٌز‬ٚ‫ اٌّؾب‬ٟ٘(‫س ِبئٍخ‬ٚ‫ِؾب‬
Co-ordinates system
‫بد‬١‫ٔظبَ ئؽذاص‬
Origin point (Origin )
ً‫ٔمطخ األط‬
178
Calculus
Deceleration
‫رجبؽإ‬
Acceleration
) ً١‫اٌزغبسع ( اٌؼغٍخ – اٌزؼغ‬
ً‫ؽغبة اٌزفبػ‬
Differential calculus
ًِ‫ػالِخ رىب‬
Sign of integration
ًِ‫ؽغبة اٌزىب‬
Integral calculus
ًِ‫اٌزىب‬ٚ ً‫ؽغبة اٌزفبػ‬
Calculus
Differentiation
ً‫رفبػ‬
Differential
ٍٟ‫رفبػ‬
ٍٟ‫ِؼبًِ رفبػ‬
Differential coefficient
‫ ٌٍّشزمبد‬ٝ‫عط‬ٌٛ‫ّخ ا‬١‫خ اٌم‬٠‫ٔظش‬
Mean value theorem for Derivatives
‫ ٌٍزىبِالد‬ٝ‫عط‬ٌٛ‫ّخ ا‬١‫خ اٌم‬٠‫ٔظش‬
Mean value theorem for Integrals
‫صبثذ‬
Constant
) ًِ‫ (ِضً صبثذ اٌزىب‬ٞ‫بس‬١‫صبثذ اخز‬
Arbitrary constant
‫لبػذح‬
Rule
Chain rule
‫لبػذح اٌغٍغٍخ‬
Constant of integration
ًِ‫صبثذ اٌزىب‬
ً‫ِزظ‬
Continuous
‫ِزظً ػٕذ ٔمطخ‬
Continuous at a point
) ً‫ش ِزظ‬١‫ِزمطغ ( غ‬
Discontinuous
Derivative
‫ِشزمخ‬
Derivation
‫اشزمبق‬
ٍٝ‫ِشزمخ ِٓ سرجخ أػ‬
Derivative of higher order
ٌٝٚ‫اٌّشزمخ األ‬
First derivative
‫خ‬١‫ِشزمخ عضئ‬
Partial derivative
ّٟٕ‫رفبػً ػ‬
Implicit differentiation
ٌٝ‫ي ئ‬ٚ‫إ‬٠
To approach
179
‫عجخ‬ِٛ ‫خ‬٠‫ب‬ٙٔ ‫ ال‬ٌٝ‫ي ئ‬ٚ‫إ‬٠
To approach plus infinity
‫خ عبٌجخ‬٠‫ب‬ٙٔ ‫ ال‬ٌٝ‫ي ئ‬ٚ‫إ‬٠
To approach minus infinity
‫ اٌظغش‬ٟ‫ ف‬ٜ ٖ‫اٌّزٕب‬
Infinitesimal
) ‫خ‬١‫بئ‬ٙٔ ‫خ ال‬١ّ‫ رىبًِ و‬ٟ‫ز‬٠‫ب‬ٙٔ ٜ‫رىبًِ ِؼزً ( ئرا وبٔذ ئؽذ‬
Improper integration
ٍٝ‫أػ‬
Upper
ٍٝ‫اٌؾذ األػ‬
Upper limit
ًِ‫ ٌٍزىب‬ٍٝ‫اٌؾذ األػ‬
Upper limit of integration
Lower limit
ٝٔ‫اٌؾذ األد‬
Integrable
ًِ‫لبثً ٌٍزىب‬
‫خ‬٠‫ب‬ٙٔ
Limit
Limit of a sequence
‫خ‬١ٌ‫خ اٌّززب‬٠‫ب‬ٙٔ
Limit on the right
ّٟٕ١ٌ‫خ ا‬٠‫ب‬ٌٕٙ‫ا‬
Limit on the left
ٜ‫غش‬١ٌ‫خ ا‬٠‫ب‬ٌٕٙ‫ا‬
‫أمالة‬
Inflexion
‫ٔمطخ أمالة‬
Inflexion point
)‫ع‬ٛ‫ٔمطخ أمالة ( ٔمطخ سع‬
Turning point
Open up wards
ٍٟ‫ػ ألػ‬ٛ‫ِفز‬
Open down wards
ً‫ػ ألعف‬ٛ‫ِفز‬
Integration
ًِ‫رىب‬
Unbounded
‫د‬ٚ‫ش ِؾذ‬١‫غ‬
‫د‬ٚ‫رىبًِ ِؾذ‬
Definite integration
‫د‬ٚ‫ش ِؾذ‬١‫رىبًِ غ‬
Indefinite integration
ٟ‫اٌزىبًِ ثبٌزغضئ‬
Integration by parts
‫غ‬٠ٛ‫اٌزىبًِ ثبٌزؼ‬
Integration by substitution
‫أؾٕبء‬
Bend ( bending )
‫ٔمطخ االٔؾٕبء‬
Bend point
Concave downwards
ً‫ِمؼش ألعف‬
Concave upwards
ٍٝ‫ِمؼش ألػ‬
‫سح‬ٚ‫د‬
Cycle
180
Cycle full
‫سح وبٍِخ‬ٚ‫د‬
Complete turn
‫سح وبٍِخ‬ٚ‫د‬
ٞ‫س‬ٚ‫د‬
Cyclic ( Periodic )
‫خ‬٠‫س‬ٚ‫ؽشوخ د‬
Periodic motion
Relative
ٝ‫ٔغج‬
Neighborhood
‫اس‬ٛ‫ع‬
Relative acceleration
ٝ‫رغبسع ٔغج‬
Differential geometry
‫خ‬١ٍ‫ٕ٘ذعخ رفبػ‬
‫فبد‬ٚ‫اٌّمز‬
Projectiles
‫فخ‬٠‫ِغبس اٌمز‬
Path of projectile
Circle
‫رّبط‬
Tangency
‫ٔمطخ اٌزّبط‬
Point of tangency
‫ِّبط‬
Tangent
‫ط‬ٚ‫ِّبط ِضد‬
Double tangent
Length of tangent
‫ي اٌّّبط‬ٛ‫ؽ‬
Chord of contact
‫رش اٌزّبط‬ٚ
Chord of a circle
‫رش اٌذائشح‬ٚ
Arc of a circle
‫ط اٌذائشح‬ٛ‫ل‬
Area of a circle
‫ِغبؽخ اٌذائشح‬
‫لطش‬
Diameter
‫لطش اٌذائشح‬
Diameter of a circle
Null circle
‫خ‬٠‫اٌذائشح اٌظفش‬
Unit circle
) ‫ؽذح‬ٌٛ‫ٔظف لطش٘ب ا‬ٚ ً‫ ِشوض٘ب ٔمطخ األط‬ٟ‫ؽذح ( اٌذائشح اٌز‬ٌٛ‫دائشح ا‬
‫اٌّّبط اٌّشزشن‬
Common tangent
ٓ١‫اٌّّبط اٌّشزشن ٌذائشر‬
Common tangent to two circles
‫ِشوض‬
Center ( centre )
‫ِشوض اٌذائشح‬
Center of a circle
181
‫ٓ ِٓ اٌخبسط‬١‫ِزّبعز‬
Touching externally
Externally touching circles
‫دائشربْ ِزّبعزبْ ِٓ اٌخبسط‬
Internally touching circles
ً‫دائشربْ ِزّبعزبْ ِٓ اٌذاخ‬
‫ٔظف لطش اٌذائشح اٌذاخٍخ‬
Inradius
ٓ١‫ٓ اٌّزجبػذر‬١‫اٌذائشر‬
Two distant circles
‫ اٌّشوض‬ٟ‫ٓ اٌّزؾذر‬١‫اٌذائشر‬
Concentric circle
ٓ١‫ٓ ِزمبؽؼز‬١‫دائشر‬
Intersecting circles
ٓ٠‫خؾ اٌّشوض‬
Line of centers
‫ٓ اٌذائشح‬١١‫رؼ‬
Identifying
‫اٌذائشح اٌذاخٍخ‬
Inscribed circle
Inscribed circle of a triangle
‫اٌذائشح اٌذاخٍخ ٌٍّضٍش‬
Escribed circle of a triangle
‫اٌذائشح اٌخبسعخ ٌٍّضٍش‬
Subtended arc
ً‫ط اٌّمبث‬ٛ‫اٌم‬
Anile(angle) of Tangency
‫خ اٌزّبط‬٠ٚ‫صا‬
‫خ‬١ٍ١‫دائشح رخ‬
Imaginary circle
Circumscribed circle of a polygon
(circumcircle)
‫طخ ثّؼٍغ‬١‫اٌذائشح اٌّؾ‬
‫لبؽغ اٌذائشح‬
Secant of a circle
Complex numbers
‫ػذد ِشوت‬
Complex number
Real part
ٟ‫م‬١‫اٌغضء اٌؾم‬
Imaginary part
ٍٟ١‫اٌغضء اٌزخ‬
ٍٟ١‫اٌزخ‬ٚ ٟ‫م‬١‫ساْ اٌؾم‬ٛ‫اٌّؾ‬
Real and imaginary axes
‫اٌغؼخ‬
Amplitude ( argument )
of a complex
)‫خ اٌؼذد اٌّشوت‬٠ٚ‫عؼخ اٌؼذد اٌّشوت ( صا‬
Absolute value of a complex number (
modulus )
‫بط اٌؼذد اٌّشوت‬١‫ِم‬
Amplitude ( argument
number(
ٓ١‫ٓ ِشوج‬٠‫ ػذد‬ٞٚ‫رغب‬
Equality of two complex numbers
182
ْ‫ػذداْ ِشوجبْ ِزشافمب‬
Conjugate complex numbers
‫خ‬١‫خ لطج‬٠ٚ‫صا‬
Polar angle
Polar coordinates
‫خ‬١‫بد لطج‬١‫ئؽذاص‬
Polar form
‫خ‬١‫سح اٌمطج‬ٛ‫اٌظ‬
‫خ ٌٍؼذد اٌّشوت‬١‫سح اٌمطج‬ٛ‫اٌظ‬
Polar form of a complex number
ٗ١‫غخ ِضٍض‬١‫ط‬
Trigonometric form
‫ٗ ٌؼذد ِشوت‬١‫غخ ِضٍض‬١‫ط‬
Trigonometric form of a complex number
Division
ُ‫مغ‬٠
Divide
Division
‫خ اٌمغّخ‬١ٍّ‫ػ‬
Divisibility
‫خ اٌمغّخ‬١ٍ‫لبث‬
ٍٝ‫مجً اٌمغّخ ػ‬٠
Divisible by
َٛ‫اٌّمغ‬
Dividend
) ُ‫ٗ (اٌمبع‬١ٍ‫َ ػ‬ٛ‫اٌّمغ‬
Divisor
‫خبسط اٌمغّخ‬
Quotient
Short division
‫شح‬١‫اٌمغّخ اٌمظ‬
Long division
‫ٍخ‬٠ٛ‫اٌمغّخ اٌط‬
‫ذ‬٠‫أعش‬
Carried out
ً‫ِٓ األفؼ‬
It is preferable
Arranging terms
‫د‬ٚ‫ت اٌؾذ‬١‫رشر‬
Descending order
ٌٟ‫ت رٕبص‬١‫رشر‬
Ascending order
ٞ‫ت رظبػذ‬١‫رشر‬
Common divisor
‫اٌمبعُ اٌّشزشن‬
ُ‫اٌمبعُ اٌّشزشن األػظ‬
Greatest common divisor
First decimal place
‫ؽبد‬٢‫ِشرجخ ا‬
Nearest ten
‫ألشة ػششح‬
Nearest unit
‫ؽذح‬ٚ ‫ألشة‬
183
Aliquot parts
) 12 ، 6 ، 4 ، 3 2 ، 1 ‫ ِضال‬12 ‫اعُ اٌؼذد‬ٛ‫اعُ ( ل‬ٛ‫اٌم‬
Approximate
‫مشة‬٠
‫ت‬٠‫اٌزمش‬
Approximation
Approximation sign
‫ت‬٠‫ػالِخ اٌزمش‬
Approximately equal
‫ت‬٠‫ ثبٌزمش‬ٞٚ‫رغب‬
Units number ( digit)
‫ؽبد‬٢‫سلُ ا‬
Denotes
ٌٝ‫ش ئ‬١‫رش‬
Perform
‫لُ ثاعشاء‬
) ُ‫ِمذاس( و‬
Magnitude
ٟ‫اٌخطأ إٌغج‬
Margin of error (relative error)
Ellipse
‫اٌمطغ إٌبلض‬
Ellipse
ٞ‫اخزالف ِشوض‬
Eccentricity
Major axis
) ‫ اٌمطغ إٌبلض‬ٟ‫س األوجش (ف‬ٛ‫اٌّؾ‬
Minor axis
)‫ اٌمطغ إٌبلض‬ٟ‫س األطغش ( ف‬ٛ‫اٌّؾ‬
Semi-major axis
‫س األوجش‬ٛ‫ٔظف اٌّؾ‬
Semi-minor axis
‫س األطغش‬ٛ‫ٔظف اٌّؾ‬
ٍٟ١‫لطغ ٔبلض رخ‬
Imaginary ellipse
ٞ‫ االخزالف اٌّشوض‬ٟ‫دائشر‬
Eccentric circle
‫ِغبؽخ اٌمطغ إٌبلض‬
Area of an ellipse
‫ٗ ٌٍمطغ إٌبلض‬١‫ع‬ٛ‫دائشح اٌز‬
Director circle of an ellipse
Ellipsoid
‫ِغغُ اٌمطغ إٌبلض‬
Center of ellipse
‫ِشوض اٌمطغ إٌبلض‬
‫خ ٌٍمطغ‬٠‫ربس اٌجإس‬ٚ‫األ‬
Focal chords of a conic
Focal radius of a conic
) ٗ١ٍ‫الؼخ ػ‬ٚ ‫ ٔمطخ‬ٞ‫أ‬ٚ ‫ٓ ثإسح اٌمطغ‬١‫اطً ث‬ٌٛ‫ّخ ا‬١‫ي اٌمطؼخ اٌّغزم‬ٛ‫ ٌٍمطغ ( ؽ‬ٞ‫اٌجؼذ اٌجإس‬
‫ثإسح اٌمطغ‬
Focus of a conic
‫اٌمطغ اٌّىبفئ‬
Parabola
184
‫اٌمطغ اٌضائذ‬
Hyperbola
Energy
‫ؽبلخ‬
Energy
Thermal energy
‫خ‬٠‫اٌطبلخ اٌؾشاس‬
Electrical energy
‫خ‬١‫شثبئ‬ٙ‫اٌطبلخ اٌى‬
Atomic energy
‫خ‬٠‫اٌطبلخ اٌزس‬
Nuclear energy
‫خ‬٠ٌٕٚٛ‫اٌطبلخ ا‬
‫خ‬١‫ى‬١ٔ‫ىب‬١ٌّ‫اٌطبلخ ا‬
Mechanical energy
Kinetic energy
‫ؽبلخ اٌؾشوخ‬
Potential energy
‫ػغ‬ٌٛ‫ؽبلخ ا‬
‫ْ ؽفع اٌطبلخ‬ٛٔ‫لب‬
Conservation of energy
‫ذ‬ٙ‫ع‬
Potential
‫ذ‬ٙ‫فشق اٌغ‬
Potential difference ( P.D)
ٓ‫ر‬ٛ١ٌٕ َ‫ْ اٌغزة اٌؼب‬ٛٔ‫لب‬
Newton's law of universal gravitation
‫ٔظف لطش اٌزغبرة‬
Gravitational radius
Stress
‫بد‬ٙ‫اإلع‬
Spiral
ٟٔٚ‫ؽٍض‬
ٓ‫عبو‬
Stationary
‫لف‬ٛ‫ٔمطخ ر‬
Stationary point
) َ‫ىب(ػٍُ ؽشوخ األعغب‬١ِ‫ٕب‬٠‫اٌذ‬
Dynamics
Kinematics
‫اٌضِبْ ثغغ إٌظش‬ٚ ْ‫ب ثبٌّىب‬ٙ‫ش ػاللز‬١‫ دساعخ ؽشوخ األعغبَ ِٓ ؽ‬ٟ‫جؾش ف‬٠ ‫ىب ( فشع‬١‫ّٕبر‬١‫اٌى‬
) َ‫ب ٘زٖ األعغب‬ِٕٙ ‫ػٓ اٌّبدح اٌّشوجخ‬
)‫ب ثبٌؾشوخ‬ٙ‫ػاللز‬ٚ ٜٛ‫ي دساعخ اٌم‬ٚ‫زٕب‬٠ ٞ‫ىب( اٌفشع اٌز‬١‫ٕبر‬١‫اٌى‬
Kinetics
Electrostatics
)‫ اٌشؾٕبد اٌغبوٕخ‬ٟ‫جؾش ف‬٠ ٞ‫شثبء اٌز‬ٙ‫خ اٌغبوٕخ( رٌه اٌفشع ِٓ اٌى‬١‫شثبئ‬ٙ‫ػٍُ اٌى‬
)‫ىب(دساعخ األعغبَ اٌغبوٕخ‬١‫ػٍُ االعزبر‬
Statics
‫اٌزّذد‬
Dilatation
Dilatation coefficient
‫ِؼبًِ اٌزّذد‬
Center of dilatation
‫ِشوض اٌزّذد‬
185
Coefficient of linear expansion
ٌٟٛ‫ِؼبًِ اٌزّذد اٌط‬
Coefficient of volume expansion
ّٟ‫ِؼبًِ اٌزّذد أٌؾغ‬
Equations
‫ِؼبدٌخ‬
Equation
Coefficient of an equation
‫ِؼبِالد اٌّؼبدٌخ‬
System of linear equations
‫خ‬١‫ٔظبَ اٌّؼبدالد اٌخط‬
‫ؽً اٌّؼبدالد‬
Solution of equations
‫خ‬١ٔ‫ِؼبدالد آ‬
Simultaneous equations
‫خ‬١‫خ خط‬١ٔ‫ِؼبدالد آ‬
Simultaneous linear equations
‫س اٌّؼبدٌخ‬ٚ‫عز‬
Roots of an equation
ٓ١ٌٛٙ‫ ٌّؼبدٌخ ثّغ‬ٟٔ‫ب‬١‫ً اٌج‬١‫اٌزّض‬
Graph of an equation in two variables
ٟٔ‫ب‬١‫اٌؾً اٌج‬
Graphical solution
ٟٕ‫ِؼبدٌخ إٌّؾ‬
Equation of a curve
‫د‬ٚ‫شح اٌؾذ‬١‫دسعخ اٌّؼبدٌخ وض‬
Degree of a polynomial equation
ٟ‫ اٌّؼبدٌخ ف‬ٟ‫ثؼشة ؽشف‬
Multiplying both sides of the equation by
Satisfies the equation
‫ؾمك اٌّؼبدٌخ‬٠
Exponential equation
‫خ‬١‫ِؼبدٌخ آع‬
Inconsistent equation
‫افمخ‬ٛ‫ش ِز‬١‫ِؼبدالد غ‬
Irrational (radical) equation
‫ش‬١‫ب اٌّزغ‬ٙ١‫ْ ف‬ٛ‫ى‬٠ ٚ‫ش رؾذ ئشبسح اٌغزس أ‬١‫ب اٌّزغ‬ٙ١‫ش ف‬ٙ‫ظ‬٠ ٟ‫ اٌّؼبدٌخ اٌز‬ٟ٘ ( ‫ِؼبدٌخ طّبء‬
3
2
x  1  0 ٚ‫أ‬
x  3  x  5 )ً‫ؼ ِض‬١‫ش طؾ‬١‫ػب ألط غ‬ٛ‫ِشف‬
Cartesian equation
‫خ‬٠‫ض‬١‫اٌّؼبدٌخ اٌىبسر‬
Fractional equation
‫خ‬٠‫ِؼبدٌخ وغش‬
‫خ‬١ٍ‫ِؼبدٌخ رفبػ‬
Differential equation
‫خ‬١‫خ خط‬١ٍ‫ِؼبدٌخ رفبػ‬
Linear differential equation
‫خ‬١‫ف‬ٛ‫ِؼبدٌخ ِظف‬
Matrix equation
‫ِؼبدٌخ اٌؾشوخ‬
Equation of motion
186
‫خ‬٠‫ِؼبدٌخ ػذد‬
Numerical equation
‫اؽذ‬ٚ ‫ي‬ٛٙ‫ِؼبدٌخ راد ِغ‬
Equation in one unknown
ٓ١ٌٛٙ‫ِؼبدٌخ راد ِغ‬
Equation in two unknowns
ٌٟٚ‫ِؼبدٌخ ِٓ اٌذسعخ األ‬
Equation of first degree
ِٓ‫ِؼبدٌخ اٌض‬
Equation of time
‫خ ِزغبٔغخ‬١ٍ‫ِؼبدٌخ رفبػ‬
Homogeneous differential equation
) ‫اؽذح‬ٚ ‫د٘ب ِٓ دسعخ‬ٚ‫غ ؽذ‬١ّ‫ْ ع‬ٛ‫ ِؼبدٌخ رى‬ٟ٘ ( ‫اٌّؼبدٌخ اٌّزغبٔغخ‬
Homogeneous equation
‫ؾ‬١‫ش لبثٍخ ٌٍزجغ‬١‫ِؼبدٌخ غ‬
Irreducible equation
‫خ‬١‫ِؼبدٌخ لطج‬
Polar equation
‫خ‬٠‫اٌّؼبدٌخ اٌغجش‬
Algebraic equation
Cubic equation
‫خ‬١‫ج‬١‫ِؼبدٌخ رىؼ‬
Linear equation
‫خ‬١‫ِؼبدٌخ خط‬
‫خ‬١ّ‫ز‬٠‫غبس‬ٌٛ ‫ِؼبدٌخ‬
Logarithmic equation
Quadratic equation
‫خ‬١‫ؼ‬١‫ِؼبدٌخ ثشث‬
Sides of an equation
‫ؽشفب اٌّؼبدٌخ‬
Degree of equation
‫دسعخ اٌّؼبدٌخ‬
‫ِؼبدٌخ ِٓ اٌذسعخ اٌخبِغخ‬
Quintic equation
Quintic polynomial
‫د ِٓ اٌذسعخ اٌخبِغخ‬ٚ‫شح ؽذ‬١‫وض‬
Equation of a circle
‫ِؼبدٌخ اٌذائشح‬
Equation of a plane
ٞٛ‫ِؼبدٌخ اٌّغز‬
Polynomial equation
‫د‬ٚ‫شح اٌؾذ‬١‫ِؼبدٌخ وض‬
Reciprocal equation
‫ثخ‬ٍٛ‫ش ثّم‬١‫ش ئرا اعزجذي اٌّزغ‬١‫اؽذ ال رزغ‬ٚ ‫ي‬ٛٙ‫ ِؼبدٌخ ثّغ‬ٟ٘ ‫خ‬١‫ِؼبدٌخ أمالث‬
Redundant equation
‫ِؼبدٌخ فبئؼخ‬
Degrees of freedom
)‫ ِؼبدٌخ ِب‬ٟ‫ رذخً ف‬ٟ‫ اٌّغزمٍخ اٌز‬ٚ‫شاد اٌؾشح أ‬١‫ ػذد اٌّزغ‬ٟ٘( ‫خ‬٠‫دسعبد اٌؾش‬
Factorization
Difference of two squares
ٓ١‫ٓ ِشثؼ‬١‫اٌفشق ث‬
Difference of two cubes
ٓ١‫ٓ ِىؼج‬١‫اٌفشق ث‬
187
ٓ١‫ع ِىؼج‬ّٛ‫ِغ‬
Sum of two cubes
ُ١‫ً ثبٌزمغ‬١ٍ‫اٌزؾ‬
Factorization by grouping
ًِ‫ِشثغ وب‬
Perfect square
‫د‬ٚ‫ صالصخ ؽذ‬ٚ‫ِشثغ وبًِ ر‬
Perfect square trinomial
)ً١ٍ‫ئوّبي اٌّشثغ ( اٌزؾ‬
Completing square
ًِ‫ا‬ٛ‫ ػ‬ٌٝ‫ؽًٍ ئ‬
Factorize
ًِ‫ا‬ٛ‫ اٌؼ‬ٌٝ‫ً ئ‬١ٍ‫اٌزؾ‬
Factoring ( factorization)
ًِ‫ا‬ٛ‫ ػ‬ٌٝ‫ً ئ‬١ٍ‫رؾ‬
Decomposition into factors
‫خ‬١ٌٚ‫اًِ أ‬ٛ‫ ػ‬ٌٝ‫ً ئ‬١ٍ‫رؾ‬
Decomposition into prime factors
Resolution
ً١ٍ‫رؾ‬
Irreducible
) ً١ٍ‫ؾ ( ٌٍزؾ‬١‫ش لبثً ٌٍزجغ‬١‫غ‬
ٍٝ‫اٌؼبًِ اٌّشزشن األػ‬
The highest common factor ( H.C.F)
ً١ٍ‫لبثً ٌٍزؾ‬
Factorable
Force
‫ح‬ٛ‫ل‬
Force
Arm of force
‫ح‬ٛ‫رساع اٌم‬
Attraction force
‫ح اٌغزة‬ٛ‫ل‬
Direction of force
‫ح‬ٛ‫ارغبٖ اٌم‬
Moment of a force
‫ح‬ٛ‫ػضَ ل‬
‫ح‬ٛ‫ش اٌم‬١‫ٔمطخ رأص‬
Point of application of force
Force of pressure
‫ح ػغؾ‬ٛ‫ل‬
Force of reaction
ً‫ح سد اٌفؼ‬ٛ‫ل‬
Composition of forces
ٜٛ‫ت اٌم‬١‫رشو‬
Equivalence of forces
ٜٛ‫رىبفإ اٌم‬
‫خ‬٠‫اص‬ٛ‫ ِز‬ٜٛ‫ل‬
Parallel forces
ٜٛ‫ أػالع اٌم‬ٞ‫اص‬ٛ‫لبػذح ِز‬
Rule of parallelogram of forces
ٞٛ‫لبػذح ِؼٍغ اٌم‬
Rule of polygon of forces
188
Friction
‫اؽزىبن‬
‫ح‬ٛ‫خؾ ػًّ اٌم‬
Line of action of a force
Centrifugal force
ٞ‫ح اٌطشد اٌّشوض‬ٛ‫ل‬
Centripetal force
ٞ‫ح اٌغزة اٌّشوض‬ٛ‫ل‬
‫ح عزة‬ٛ‫ل‬
Attraction force
‫ح اؽزىبن‬ٛ‫ل‬
Friction force
‫ِؼبًِ االؽزىبن‬
Coefficient of friction
ٜٛ‫لبػذح ِضٍش اٌم‬
Rule of triangle of forces
َ‫لذ‬
Foot
Foot – pound
‫ه ٔمطخ‬٠‫ رؾش‬ٟ‫اؽذ ف‬ٚ ‫ٔذ‬ٚ‫ح ِمذاس٘ب ثب‬ٛ‫ رٕزغٗ ل‬ٞ‫ ِمذاس اٌشغً اٌز‬ٝ٘ٚ ً‫ؽذح شغ‬ٚ ( َ‫ٔذ لذ‬ٚ‫ثب‬
) ‫اؽذح‬ٚ َ‫ش٘ب ِغبفخ لذس٘ب لذ‬١‫رأص‬
Foot- poundal
َ‫ٔذاي لذ‬ٚ‫ثب‬
Harmonic motion
) ‫ ط ِمذاس اإلصاؽخ‬، ‫ أ صبثذ‬، ‫ح‬ٛ‫ش ق اٌم‬١‫ أ ط ؽ‬- = ‫خ )ق‬١‫افم‬ٛ‫اٌؾشوخ اٌز‬
ٟ‫افم‬ٛ‫عؾ اٌز‬ٌٛ‫ا‬
Harmonic average (mean)
Relativity
‫خ‬١‫خ إٌغج‬٠‫ٔظش‬
Inertia
ٟ‫س اٌزار‬ٛ‫اٌمظ‬
ٟ‫س اٌزار‬ٛ‫ػضَ اٌمظ‬
Moment of inertia
‫ اٌؾشوخ‬ٟ‫رٓ ف‬ٛ١ٔ ٓ١ٔ‫ا‬ٛ‫ل‬
Newton's laws of motion
Parallel forces
‫خ‬٠‫اص‬ٛ‫ ِز‬ٜٛ‫ل‬
Resolution of forces
ٜٛ‫ً اٌم‬١ٍ‫رؾ‬
‫لذسح‬
Power
‫خ‬١‫ٕ٘ذعخ ئؽذاص‬
Co-ordinate geometry
‫خ اٌؾشوخ‬١ّ‫و‬
Momentum
‫ؽشوخ‬
Motion
‫ؽشوخ ِٕزظّخ‬
Uniform motion
Weigh
ْ‫ض‬٠
Weight
ْ‫ص‬ٚ , ً‫صم‬
Centre of gravity
ً‫ِشوض اٌضم‬
Centre of mass
‫ِشوض اٌىزٍخ‬
189
َ‫اٌؼض‬
Moment
Centre of moments
َ‫ِشوض اٌؼض‬
Moment of couple
‫اط‬ٚ‫ػضَ االصد‬
‫ت‬١‫لؼ‬
Rod
Fractions
ٞ‫بد‬١‫وغش اػز‬
Vulgar fraction
Common fraction
ٜ‫وغش ػبد‬
Compound fraction
‫وغش ِشوت‬
ٍٟ‫وغش فؼ‬
Proper fraction e.g. 2/5
ٞ‫وغش ِغبص‬
Improper fraction e.g. 7/3
ٟ‫وغش عضئ‬
Partial fraction
‫رغضئخ اٌىغش‬
Decomposition of a fraction
) 0.2, 0.10( ٞ‫وغش ػشش‬
Decimal fraction
‫خ‬٠‫فبطٍخ ػشش‬
Decimal point
Mixed decimal
) 38.21 ً‫ ِض‬ٞ‫وغش ػشش‬ٚ ‫ؼ‬١‫ْ ِٓ ػذد طؾ‬ٛ‫زى‬٠ ٞ‫ ػذد ػشش‬ٛ٘ (‫ ِشوت‬ٞ‫وغش ػشش‬
‫اخزضاي اٌىغش‬
Reduction of a fraction
Repeating decimal(circulating decimal= recurring decimal )
ٞ‫س‬ٚ‫وغش د‬
ٞ‫وغش عجش‬
Algebraic fraction
) 1 = ‫ؽذح ( اٌجغؾ‬ٌٛ‫وغش ا‬
Unit fraction
‫ِمبَ اٌىغش‬
Denominator of a fraction
ٜ‫وغش‬
Fractional
ٜ‫ػذد وغش‬
Mixed number
‫س‬ٛ‫اٌزخٍض ِٓ اٌىغ‬
Clearing of fractions
‫خ‬١ٌ‫ّخ إٌّض‬١‫اٌم‬
Place value
ٌٝٚ‫خ األ‬٠‫إٌّضٌخ اٌؼشش‬
First decimal place
ٗٙ‫س ِزشبث‬ٛ‫وغ‬
Similar fractions
‫س اٌّزىبفئخ‬ٛ‫اٌىغ‬
Equivalent fractions
190
‫س‬ٛ‫ت اٌىغ‬١‫رشر‬
Ordering fractions
Functions
Analytic function
ٍٟ١ٍ‫الزشاْ رؾ‬
Function
) ‫الزشاْ ( داٌخ‬
‫الزشاْ ِشوت‬
Composite function
‫الزشاْ صبثذ‬
Constant function
)‫الزشاْ ِزظً (ِغزّش‬
Continuous function
‫ِزٕبلض‬
Decreasing
Decreasing function
‫الزشاْ ِزٕبلض‬
Increasing function
‫ذ‬٠‫الزشاْ ِزضا‬
ًِ‫ش شب‬١‫الزشاْ غ‬
Into function
ٟ‫االلزشاْ اٌؼىغ‬
Inverse function
) ٓ٠‫اؽذ ( ِزجب‬ٌٛ ‫اؽذ‬ٚ ْ‫الزشا‬
One-to-one(injection) function
)ٜ‫الزشاْ شبًِ(اٌّغبي اٌّمبثً=اٌّذ‬
Onto (surjection ) function
‫الزشاْ اٌزٕبظش‬
One - to - one (biJection) function
Composition of functions
‫ت االلزشأبد‬١‫رشو‬
Equality of functions
‫ االلزشأبد‬ٜٚ‫رغب‬
Monotonic functions
‫ ال‬ٟ‫ اٌّزٕبلظخ اٌز‬ٚ‫ ال رزٕبلض أثذا أ‬ٟ‫ذح اٌز‬٠‫ االلزشأبد اٌّزضا‬ٟ٘ ) )‫ٗ (ِطشدح‬٠‫ش‬١‫ر‬ٚ ‫الزشأبد‬
) ‫ذ أثذا‬٠‫رزضا‬
) ‫خ‬٠‫ٗ ( دائش‬١‫الزشأبد ِضٍض‬
Trigonometric functions
Zeros of function
‫أطفبس اٌذاٌخ‬
Constant function
‫اٌذاٌخ اٌضبثزخ‬
‫خ‬٠‫اٌذاٌخ اٌىغش‬
Fractional function
‫ اٌذاٌخ‬ٞ‫ِذ‬
Range of function
‫اي‬ٚ‫اؽشاد اٌذ‬
Monotony of functions
Density function
‫داٌخ اٌىضبفخ‬
Even function
‫خ‬١‫ع‬ٚ‫داٌخ ص‬
Odd function
‫خ‬٠‫داٌخ فشد‬
191
‫خ‬١‫اي اٌّضٍض‬ٚ‫اٌذ‬
Trigonometric function
Function of a function
‫داٌخ اٌذاٌخ‬
Derivative function
‫داٌخ ِشزمخ‬
Limit of a function
‫خ اٌذاٌخ‬٠‫ب‬ٙٔ
‫ؾخ‬٠‫داٌخ طش‬
Explicit function
‫خ‬١‫داٌخ آع‬
Exponential function
Reciprocal function
‫خ‬١‫داٌخ ػىغ‬
Integrable function
ًِ‫داٌخ لبثٍخ ٌٍزىب‬
‫سح ػٕظش‬ٛ‫ط‬
Image of an element
‫ِغبي‬
Domain
‫ِغبي اٌذاٌخ‬
Domain of a function
‫اٌّغبي اٌّشزشن‬
Common domain
Co-domain
ً‫اٌّغبي اٌّمبث‬
Field of definition
‫ف‬٠‫ِغبي اٌزؼش‬
Geometry
Triangle
‫اٌّضٍش‬
Square
‫اٌّشثغ‬
‫ِشثؼبد ِزذاخٍخ‬
Interfered squares
Rhombus
ٓ١‫ايِؼ‬
Rectangle
ً١‫اٌّغزط‬
) ‫لبئُ ( ِزؼبِذ‬
Rectangular
‫شجٗ ِٕؾشف‬
Trapezoid ( trapezium )
‫لبػذح شجٗ إٌّؾشف‬
Trapezoidal rule
Median of trapezium (or trapezoid)
‫ ٌشجٗ إٌّؾشف‬ٝ‫عط‬ٚ ‫لبػذح‬
Isosceles trapezoid
ٓ١‫ اٌغبل‬ٞٚ‫شجٗ ِٕؾشف ِزغب‬
‫ أػالع‬ٞ‫اص‬ٛ‫ِز‬
Parallelogram
ٟ‫ِؼٍغ خّبع‬
Pentagon
192
Hexagon
ٟ‫اٌغذاع‬
Heptagon
ٟ‫اٌغجبػ‬
Octagon
ٟٔ‫اٌضّب‬
Nonagon
ٟ‫اٌزغبػ‬
Decagon
ٞ‫اٌؼشبس‬
‫دائشح‬
Circle
‫ٔظف دائشح‬
Semi circle
Curve
ٕٝ‫ِٕؾ‬
Polygon
‫ِؼٍغ‬
ُ‫ِؼٍغ ِٕزظ‬
Regular polygon
ُ‫ِشوض ِؼٍغ ِٕزظ‬
Center of a regular polygon
‫لطش اٌّؼٍغ‬
Diagonal of a polygon
‫اسرفبع اٌّضٍش‬
Altitude of a triangle
ًّ‫ئٔشبء ػ‬
Construction
ٟ‫ٕذع‬ٌٙ‫لبػذح اٌشىً ا‬
Base
‫ اٌّضٍش‬ٟ‫زب اٌمبػذح ف‬٠ٚ‫صا‬
Base angles of a triangle
‫ػٍغ‬
Side
‫ػٍغ ِشزشن‬
Common side
‫خ‬٠‫ٔظش‬
Theorem
‫خ‬٠‫ػىظ إٌظش‬
Converse of a theorem
‫سثغ دائشح‬
Quadrant of a circle
ٟ‫شىً سثبػ‬
Quadrilateral
ٞ‫ دائش‬ٟ‫سثبػ‬
Circular quadrilateral (cyclic)
Subtend
)‫خ‬٠‫خ اٌّشوض‬٠ٚ‫مبثً اٌضا‬٠ ‫ط اٌذائشح‬ٛ‫ ل‬،‫خ‬ٙ‫اع‬ٌّٛ‫خ ا‬٠ٚ‫مبثً اٌضا‬٠ ‫مبثً ( ػٍغ اٌّضٍش‬٠
Similar figures
‫خ‬ٙ‫أشىبي ِزشبث‬
Similar triangles
‫خ‬ٙ‫ِضٍضبد ِزشبث‬
‫ؾ‬١‫ٔظف ِؾ‬
Semi -circumference
) ‫بٖ لبئّخ‬٠‫ا‬ٚ‫ ِٓ ص‬ٞ‫ْ أ‬ٛ‫ ال رى‬ٞ‫ِضٍش ِبئً ) اٌّضٍش اٌز‬
Oblique triangle
) ‫ ٔمطخ اٌزمبء اسرفبػبد اٌشىً وبٌّضٍش ِضال‬ٛ٘ (‫ االسرفبػبد‬ٝ‫ٍِزم‬
Orthocenter
193
‫اع اٌّضٍش‬ٛٔ‫أ‬
Types of triangle
Right-angled triangle
‫خ‬٠ٚ‫ِضٍش لبئُ اٌضا‬
Hypotenuse
ُ‫رش اٌّضٍش اٌمبئ‬ٚ
Acute - angled triangle
‫ب‬٠‫ا‬ٚ‫ِضٍش ؽبد اٌض‬
‫ األػالع‬ٞٚ‫ِضٍش ِزغب‬
Equilateral triangle
Isosceles triangle
ٓ١‫ اٌغبل‬ٞٚ‫ِضٍش ِزغب‬
Scalene triangle
‫ِضٍش ِخزٍف األػالع‬
) ‫بد‬١‫بٔبد ( ِؼط‬١‫ث‬
Data (Given)
‫ٕظف‬٠
Bisect
ٗ‫رشبث‬
Similitude (similarity)
Center of similitude
ٗ‫ِشوض اٌزشبث‬
Ratio of similitude
ٗ‫ٔغجخ اٌزشبث‬
‫ؾ‬١‫اٌّؾ‬
Perimeter
Circumference
‫ؾ اٌذائشح‬١‫ِؾ‬
Plane geometry
‫خ‬٠ٛ‫ٕ٘ذعخ ِغز‬
‫ٕذعخ‬ٌٙ‫ا‬
Geometry
Area
‫ِغبؽخ‬
‫اٌّغبؽخ اٌّظٍٍخ‬
Shaded area
‫ي‬ٛ‫اٌط‬
Length
Breadth ( width )
‫اٌؼشع‬
Height
‫االسرفبع‬
‫خ‬١ٍ١ٍ‫ٕذعخ اٌزؾ‬ٌٙ‫ا‬
Analytic geometry
‫خ‬١‫م‬١‫بد اٌزطج‬١‫بػ‬٠‫اٌش‬
Applied mathematics
Mathematics of finance
‫خ‬١ٌ‫بد اٌّب‬١‫بػ‬٠‫اٌش‬
Pure mathematics
‫بد اٌجؾزخ‬١‫بػ‬٠‫اٌش‬
‫ِمذِخ‬
Foreword
Perpendicular bisector
ٞ‫د‬ّٛ‫إٌّظف اٌؼ‬
Geometric instrument
‫خ‬١‫ٍخ ) ٕ٘ذع‬١‫ع‬ٚ( ‫أداح‬
194
A set – square
) ‫خ‬١‫( ِضٍش لبئُ – أداح ٕ٘ذع‬
‫أٔظبف ألطبس‬
Radii
‫ِٕمٍخ‬
Protractor
ً‫عٓ اٌجشع‬
Pin ( sharp point )
) ‫ؾ‬١‫ؽجً ( خ‬
Thread
‫ؾ‬٠‫شش‬
Tape
‫إٌّطمخ اٌّظٍٍخ‬
Shaded region
Scissors
‫ِمض‬
Generalization
ُ١ّ‫اٌزؼ‬
Median of a triangle
‫عؾ اٌّضٍش‬ٛ‫ِز‬
Opposite figure
ً‫اٌشىً اٌّمبث‬
‫ِغأٌخ‬
Problem
Deduce that
ْ‫رغزٕزظ أ‬
Longest side
‫اوجش ػٍغ‬
Compasses
‫فشعبس‬
Conclusion
‫اعزٕزبط‬
ْ‫اٌجش٘ب‬
Proof
Fixed point
‫ٔمطخ صبثزخ‬
Point of division
ُ١‫ٔمطخ رمغ‬
‫ٔمطخ خبسعخ‬
External point
‫ُ ِٓ اٌخبسط‬١‫اٌزمغ‬
External division
‫لطؼخ‬
Segment
ٟٕ‫لطؼخ ِٓ ِٕؾ‬
Segment of a curve
‫ّخ‬١‫لطؼخ ِغزم‬
Line segment
Midpoint of a line segment
‫ّخ‬١‫ف لطؼخ ِغزم‬١‫رٕظ‬
Length of a line segment
‫ّخ‬١‫ي اٌمطؼخ اٌّغزم‬ٛ‫ؽ‬
‫رش‬ٚ
Chord
ٟٕ‫رش إٌّؾ‬ٚ
Chord of a curve
) ‫ط‬ٚ‫سأط ( سؤ‬
Vertex ( vertices )
195
Consecutive vertices
‫خ‬١ٌ‫ط ِززب‬ٚ‫سؤ‬
Vertices of a triangle
‫ط اٌّضٍش‬ٚ‫سؤ‬
ٍٝ‫ ػ‬ٞ‫د‬ّٛ‫ػ‬
Perpendicular to
ٓ٠‫ِزؼبِذ‬
Orthogonal
‫اؽذح‬ٚ ‫ اعزمبِخ‬ٍٝ‫ػ‬
Collinear
Collinear points
‫اؽذح‬ٚ ‫ اعزمبِخ‬ٍٝ‫ٔمؾ ػ‬
Non-collinear
‫اؽذح‬ٚ ‫ اعزمبِخ‬ٍٝ‫ا ػ‬ٛ‫غ‬١ٌ
‫سأط لّخ اٌّضٍش‬
Apeso
‫ٔمطخ رمبؽغ االسرفبػبد ٌٍّضٍش‬
Ortho Centre
‫عطبد اٌّضٍش‬ٛ‫ِز‬
Medians of the Triangles
Convex Polygon
‫ِؼٍغ ِؾذة‬
Concave polygon
‫ِؼٍغ ِمؼش‬
Convex curve
‫ ِؾذة‬ٕٝ‫ِٕؾ‬
Concave curve
‫ ِمؼش‬ٟٕ‫ِٕؾ‬
‫ة‬ٍٛ‫اٌّط‬
Required to Prove (R.T.P)
َٛٙ‫ِف‬
Concept
‫ف‬٠‫رؼش‬
Definition
‫أثؼبد‬
Dimensions
‫ثبٌؼىظ‬ٚ
And Conversely
‫سح‬ٛ‫ط‬
Image
‫ب‬٠‫ا‬ٚ‫ اٌض‬ٞٚ‫ِزغب‬
Equiangular
Equiangular figures
‫ب‬٠‫ا‬ٚ‫خ اٌض‬٠ٚ‫أشىبي ِزغب‬
Equiangular polygon
‫ب‬٠‫ا‬ٚ‫ اٌض‬ٞٚ‫ِؼٍغ ِزغب‬
Equiangular triangle
‫ب‬٠‫ا‬ٚ‫ اٌض‬ٞٚ‫ِضٍش ِزغب‬
‫ األػالع‬ٞٚ‫ِزغب‬
Equilateral
‫ األػالع‬ٞٚ‫ِؼٍغ ِزغب‬
Equilateral polygon
Translation
‫االٔزمبي‬
Rotation
ْ‫سا‬ٚ‫د‬
ْ‫سا‬ٚ‫ِشوض اٌذ‬
Center of rotation
196
‫أؼىبط‬
Reflection
ُ١‫ِغزم‬ٚ ‫ٓ ٔمطخ‬١‫اٌجؼذ ث‬
Distance from a point to a line
Distance between two parallel lines
ٓ١٠‫اص‬ٛ‫ٓ ِز‬١ّ١‫ٓ ِغزم‬١‫اٌجؼذ ث‬
Distance between two parallel planes
ٓ١٠‫اص‬ٛ‫ٓ ِز‬١٠ٛ‫ٓ ِغز‬١‫اٌجؼذ ث‬
ٜٛ‫ِغز‬ٚ ‫ٓ ٔمطخ‬١‫اٌجؼذ ث‬
Distance from a point to a plane
‫رطبثك‬
Congruence
‫رطبثك األشىبي‬
Congruence of figures
‫ِزطبثك‬
Congruent
Congruent polygons
‫ِؼٍؼبد ِزطبثمخ‬
Congruent triangles
‫ِضٍضبد ِزطبثمخ‬
Identical(Congruent) figures
‫أشىبي ِزطبثمخ‬
Coincident configurations
‫أشىبي ِزطبثمخ‬
‫ِطبثمخ‬
Identification
Identify
‫ؽبثك‬
Identity
‫ ِطبثمخ‬، ‫رطبثك‬
‫ِطبثك ٌـ‬
Identical
ٛ‫عطؼ ِغز‬
Flat surface plane
ٌٟ‫ِضب‬
Ideal
Ideal points
‫خ‬١ٌ‫ٔمؾ ِضب‬
If and only if
‫فمؾ ئرا‬ٚ ‫ئرا‬
If …….then
ْ‫ فا‬....... ‫ئرا‬
‫ِؾبؽ‬
Inscribed
‫دائشح ِؾبؽخ‬
Inscribed circle
Inscribed circle of a polygon
‫دائشح ِؾبؽخ ثّؼٍغ‬
Inscribed triangle of a circle
‫ِضٍش ِؾبؽ ثذائشح‬
‫خ‬٠‫اص‬ٛ‫أػالع ِز‬
Parallel sides
Hexagonal
‫ب‬٠‫ا‬ٚ‫اٌض‬ٚ ‫ األػالع‬ٟ‫عذاع‬
Hexahedral
‫ػ‬ٛ‫ اٌغط‬ٟ‫عذاع‬
197
Lamina
‫ؾخ‬١‫طف‬
Shrinking
‫ش‬١‫رظغ‬
‫ش‬١‫رىج‬
Magnification (Enlargement )
Magnification center
‫ش‬١‫ِشوض اٌزىج‬
Magnification ratio
‫ش‬١‫ِؼبًِ اٌزىج‬
ٟ‫ٕ٘ذع‬
Geometric
Groups
‫صِشح‬
Group
Commutative group
ٗ١ٍ٠‫صِشح رجذ‬
Commutative law
ً٠‫ْ اٌزجذ‬ٛٔ‫لب‬
Composite group
‫صِشح ِشوجخ‬
Cyclic group
‫خ‬٠‫س‬ٚ‫صِشح د‬
Finite group
‫خ‬١ٙ‫صِشح ِٕز‬
‫خ‬١ٙ‫ش ِٕز‬١‫صِشح غ‬
Infinite group
Permutation group
ً٠‫صِشح اٌزجبد‬
Simple group
‫طخ‬١‫صِشح ثغ‬
Sub-group
‫خ‬١‫صِشح عضئ‬
) ٗ‫ٔظف ( شج‬
Semi
‫شجٗ صِشح‬
Semi- group
Interest
‫فبئذح‬
Interest
Rate of interest
‫عؼش اٌفبئذح‬
Compound interest
‫فبئذح ِشوجخ‬
‫طبف‬
Net
ٟ‫اٌشثؼ اٌظبف‬
Net profit
‫ِؼذي‬
Rate
198
) ‫ٓ ( لشع‬٠‫ّد‬
Debt
‫ظؾؼ‬٠
Debug
Decade
) ‫اد‬ٕٛ‫ػمذ ( ػشش ع‬
Receipt
)‫ظبي‬٠‫طً اعزالَ ( ئ‬ٚ
‫خ‬١ٌ‫ّخ اٌؾب‬١‫اٌم‬
Present value
‫خ‬٠ٕٛ‫خ ٌٍذفغ اٌغ‬١ٌ‫ّخ اٌؾب‬١‫اٌم‬
Present value of an annuity
‫خ‬٠ٕٛ‫خ ع‬١ٌ‫دفؼخ ِب‬
Annuity
‫ّخ‬٠‫ ِغزذ‬ٚ‫اطٍخ أ‬ٛ‫خ ِز‬١ٌ‫دفؼخ ِب‬
Continued annuity
Annuity contract
‫ػمذ اٌذفغ‬
Deferred annuity
‫خ ِإعٍخ‬١ٌ‫دفؼخ ِب‬
Ordinary annuity
‫خ‬٠‫خ ػبد‬١ٌ‫دفؼخ ِب‬
Perpetual annuity
‫ّخ‬٠‫دفغ ِغزذ‬
Semi – annual
ٕٞٛ‫ٔظف ع‬
Broker
‫عّغبس‬
Brokerage
‫عّغشح‬
Instalment
) ‫ؾ‬١‫لغؾ ( رمغ‬
‫ألغبؽ‬
Instalments
) ‫سح‬ٛ‫بٌخ ( فبر‬١‫وّج‬
Bill
‫ذاػبد‬٠‫اإل‬
Deposits
‫اعزجذاي اٌؼٍّخ‬
Exchange of money
‫ربعش‬
Merchant
‫الد‬ٚ‫ِؾب‬
Trials
Lines
ُ١‫ِغزم‬
Straight
ُ١‫خؾ ِغزم‬
Straight line
ً١ٌّ‫ا‬
Slope ( gradient )
ُ١‫ً اٌّغزم‬١ِ
Slope of a straight line
199
‫ِزؼبِذ‬
Orthogonal
ٓ١٠‫اص‬ٛ‫ٓ ِز‬١ّ١‫ٓ ِغزم‬١‫ ث‬ٞ‫د‬ّٛ‫اٌجؼذ اٌؼ‬
Distance between two parallel lines
ٍٝ‫د إٌبصي ِٓ إٌمطخ ػ‬ّٛ‫ي اٌؼ‬ٛ‫ ؽ‬ٛ٘ (ٞٛ‫ِغز‬ٚ ‫ٓ ٔمطخ‬١‫اٌجؼذ ث‬
Distance from a point to plane
)ٜٛ‫اٌّغز‬
ُ١‫ِغزم‬ٚ ‫ٓ ٔمطخ‬١‫اٌجؼذ ث‬
Distance from a point to a line
‫ٔظف‬
Half
)ُ١‫شؼبع ( ٔظف ِغزم‬
Half line=RAY
Curved Line
ٕٝ‫خؾ ِٕؾ‬
Line Symmetry
ً‫خؾ اٌزّبص‬
ٖ‫ارغب‬
Direction
ُ١‫ارغبٖ ِغزم‬
Direction of a line
ٝ‫ش خط‬١‫غ‬
Non- linear
ُ١‫ خؾ ِغزم‬ٟ‫االٔؼىبط ف‬
Reflection in a line
Parallel lines
‫خ‬٠‫اص‬ٛ‫ّبد ِز‬١‫ِغزم‬
Perpendicular lines
‫ّبد ِزؼبِذح‬١‫ِغزم‬
Oblique lines
‫ّبد ِبئٍخ‬١‫ِغزم‬
Perpendicular
ٞ‫د‬ّٛ‫ػ‬
Common perpendicular
‫ اٌّشزشن‬ٞ‫د‬ّٛ‫اٌؼ‬
Perpendicular to a plane
ٛ‫ ِغز‬ٍٝ‫ ػ‬ٞ‫د‬ّٛ‫ُ اٌؼ‬١‫اٌّغزم‬
ٌٟ‫خؾ ِضب‬
Ideal line
Parallel lines
‫خ‬٠‫اص‬ٛ‫ؽ ِز‬ٛ‫خط‬
Concurrent lines
‫خ‬١‫ؽ ِزالل‬ٛ‫خط‬
ً‫خؾ ِبئ‬
Inclined line
‫خؾ ِٕىغش‬
Broken line
) ُ١‫لبؽغ ( خؾ ِغزم‬
Secant
Transversal
) ‫ُ لبؽغ‬١‫لبؽغ ( ِغزم‬
Vertical line
) ٟ‫ ( سأع‬ٞ‫د‬ّٛ‫خؾ ػ‬
ٟ‫خؾ اثزذائ‬
Initial line
‫اء‬ٛ‫خؾ االعز‬
Equator
200
Matrices
‫فخ‬ٛ‫ِظف‬
Matrix
‫فخ ِشثؼخ‬ٛ‫ِظف‬
Square matrix
‫ أػّذح‬ٌٝ‫ف ئ‬ٛ‫ً اٌظف‬٠ٛ‫فخ (رؾ‬ٛ‫ي اٌّظف‬ٛ‫ِٕم‬
Transpose of a matrix
)‫اٌؼىظ‬ٚ
Rank of matrix
‫فخ‬ٛ‫سرجخ اٌّظف‬
A djoint matrix
‫فخ ِشافمخ‬ٛ‫ِظف‬
‫ِٕفشدح‬
Singular
Non – Singular
‫ش ِٕفشدح‬١‫غ‬
‫فخ اٌّؼبِالد‬ٛ‫ِظف‬
Matrix of coefficients
)‫فبد اٌّشثؼخ‬ٛ‫ اٌّظف‬ٟ‫اٌؼبًِ اٌّشافك( ف‬
Cofactor
‫فخ‬ٛ‫ط ) اٌّظف‬ٛ‫ش ( ِؼى‬١‫ٔظ‬
Inverse of matrix
‫سرجخ‬
Rank (Order )
ٟ‫غ‬١‫اٌمطش اٌشئ‬
Principal or leading diagonal
‫خش‬٢‫اٌمطش ا‬
Secondary diagonal
Conjugate diameters
ْ‫لطشاْ ِزشافمب‬
Conjugate elements
‫اٌؼٕبطش اٌّزشافمخ‬
Names
‫ِخطؾ ( شىً ) أسعب ٔذ‬
Argand diagram
De Moivre's formula
‫فش‬ِٛ ٞ‫ْ د‬ٛٔ‫لب‬
Apollonlus' theorem
‫ط‬ٛ١ٌٔٛ ٛ‫خ أث‬٠‫ٔظش‬
ْ‫شِب‬١‫عج‬
Spearman
ْ‫ ِشعب‬ٞ‫ٔب د‬ٛٔ‫لب‬
De Morgan laws
ٓ‫ِخطؾ ف‬
Venn diagram
Pythagoras' theorem
‫سس‬ٛ‫ضبغ‬١‫خ ف‬٠‫ٔظش‬
Ptolemy's theorem
‫ط‬ّٛ١ٍ‫خ ثط‬٠‫ٔظش‬
201
‫لبػذح وشاِش‬
Cramer's rule
ِٟ‫لبػذح ال‬
Lami's rule
ٓ٠‫س‬ٍٛ‫ِزغٍغٍخ ِبو‬
Maclaurin's series
Taylor's series
‫س‬ٍٛ١‫ِزغٍغٍخ ر‬
Pascal triangle
‫ِضٍش ثبعىبي‬
‫ِغٍّخ‬
Postulate
‫ذط‬١ٍ‫ِغٍّخ ئل‬
Postulate of Euclid
Numbers
‫خ‬١‫ؼ‬١‫األػذاد اٌطج‬
Natural numbers
Sum of cubes of natural numbers
‫خ‬١‫ؼ‬١‫ع ِىؼجبد األػذاد اٌطج‬ّٛ‫ِغ‬
Sum of squares of natural numbers
‫خ‬١‫ؼ‬١‫ع ِشثؼبد األػذاد اٌطج‬ّٛ‫ِغ‬
‫ؾخ ِزطبثمخ‬١‫أػذاد طؾ‬
Congruent integers
‫خ‬١‫ش إٌغج‬١‫أالػذاد غ‬
Irrational numbers
Rational numbers
‫خ‬١‫أالػذاد إٌغج‬
Real numbers
‫خ‬١‫م‬١‫األػذاد اٌؾم‬
‫خ‬١‫م‬١‫ػشة األػذاد اٌؾم‬
Product of real numbers
‫خ‬١ٍ١‫األػذاد اٌزخ‬
Imaginary numbers
‫األػذاد اٌّخزٍطخ‬
Mixed numbers
‫بط األػذاد‬١‫ِم‬
Number scale
ٚ‫ب ئِب عبٌجخ أ‬ٙ‫ ئشبسار‬ٟ‫خ ( األػذاد اٌز‬ٙ‫أػذاد ِزغ‬
Directed numbers
)‫عجخ‬ِٛ
‫خ‬٠‫األػذاد اٌغجش‬
Algebraic numbers
ٟ‫ع‬ٚ‫ص‬
Even
ٟ‫ع‬ٚ‫ػذد ص‬
Even number
ٞ‫فشد‬
Odd
ٞ‫ػذد فشد‬
Odd number
Number theory
‫خ األػذاد‬٠‫ٔظش‬
Arabic numerals
‫خ‬٠‫ٕذ‬ٌٙ‫األسلبَ ا‬
202
‫ؼ‬١‫ػذد طؾ‬
Whole number
) ‫ىب ( ػششح أػؼبف‬٠‫د‬
Deca
) ‫اد‬ٕٛ‫ػمذ( ػشش ع‬
Decade
‫ة ػذد‬ٍٛ‫ِم‬
Reciprocal of a number
ٌٟٚ‫ػذد أ‬
Prime number
ٌٟٚ‫ش أ‬١‫ػذد غ‬
Composite number
Inverse of a number
‫ة اٌؼذد‬ٍٛ‫ِم‬
Whole number
‫ؼ‬١‫ػذد طؾ‬
Number of units
‫ؽذاد‬ٌٛ‫ػذد ا‬
‫ٔفظ اٌؼذد‬
Same number
‫ؼ‬١‫ػذد طؾ‬
Integer
‫عبٌت‬
Negative
‫ؼ عبٌت‬١‫ػذد طؾ‬
Negative integer
‫عت‬ِٛ
Positive
‫عت‬ِٛ ‫ؼ‬١‫ػذد طؾ‬
Positive integer
Cube of a number
‫ِىؼت اٌؼذد‬
Absolute number
)‫ض‬١ِّ ‫ش‬١‫ غ‬ّٟ‫اٌؼذد اٌّطٍك ( ػذد سل‬
‫ؼ‬١‫اؽذ اٌظؾ‬ٌٛ‫ا‬
Unity
Ratio
‫ٔغجخ‬
Ratio
‫ِزٕبعت‬
‫ح‬
Proportional
‫رٕبعت‬
Proportion
‫ؽذا ٔغجخ‬
Terms of a ratio
Antecedent
) َ‫ي ( اٌّمذ‬ٚ‫اٌؾذ األ‬
Consequent
) ٌٟ‫ ( اٌزب‬ٟٔ‫اٌؾذ اٌضب‬
‫خ‬٠ٛ‫ٔغجخ ِئ‬
Percent
Numerator (Top)
‫اٌجغؾ‬
Denominator
َ‫اٌّمب‬
203
Reduction of fractions to a common
denominator
‫ذ اٌّمبِبد‬١‫ؽ‬ٛ‫ر‬
Fourth proportional
‫اٌشاثغ اٌّزٕبعت‬
Third proportional
‫اٌضبٌش اٌّزٕبعت‬
Mean proportional
‫عؾ اٌّزٕبعت‬ٌٛ‫ا‬
Extremes of proportion
‫ اٌزٕبعت‬ٟ‫ؽشف‬
Extreme
‫ؽشف اٌزٕبعت‬
Means of proportion
‫ اٌزٕبعت‬ٟ‫عط‬ٚ
Continued proportion
ً‫اٌزٕبعت اٌّزغٍغ‬
Proportional division
ٟ‫ُ اٌزٕبعج‬١‫اٌزمغ‬
Proportional quantities
‫بد ِزٕبعجخ‬١ّ‫و‬
‫صبثذ اٌزٕبعت‬
Factor of proportionality
‫ؽذ رٕبعت‬
Term of proportion
‫ش‬١‫ِؼذي اٌزغ‬
Rate of change
Directly proportional
ٞ‫رٕبعت ؽشد‬
Inversely proportional
ٟ‫رٕبعت ػىغ‬
Direct variation
ٞ‫ش ؽشد‬١‫رغ‬
Inverse variation
ٟ‫ش ػىغ‬١‫رغ‬
sequences
) ‫خ‬١ٌ‫ا‬ٛ‫خ ( ِز‬١ٌ‫ِززب‬
Sequence( progression)
‫خ‬١ٌ‫ؽذ ِززب‬
Term of sequence
‫خ‬١‫خ ) ؽغبث‬١ٌ‫ا‬ٛ‫خ ( ِز‬١ٌ‫ِززب‬
Arithmetic progression (sequence)
‫خ‬١‫خ اٌؾغبث‬١ٌ‫أعبط اٌّززب‬
Common difference in an arithmetic progression
) ٞ‫عؾ اٌؼذد‬ٌٛ‫ (ا‬ٟ‫عؾ اٌؾغبث‬ٛ‫اٌّز‬
Arithmetic mean
‫ِزغٍغٍخ‬
Series
‫خ‬١‫ِزغٍغٍخ ؽغبث‬
Arithmetic series
Term of a series
‫ؽذ ِزغٍغٍخ‬
Geometric series
‫خ‬١‫ِزغٍغٍخ ٕ٘ذع‬
204
‫خ ِزٕبلظخ‬١‫ِزغٍغٍخ ٕ٘ذع‬
Decreasing geometrical series
‫خ‬١‫ِزغٍغٍخ آع‬
Exponential series
Finite series
‫خ‬١ٙ‫ِزغٍغٍخ ِٕز‬
Geometric progression
‫خ‬١‫خ ٕ٘ذع‬١ٌ‫ِززب‬
‫خ‬١‫ٕذع‬ٌٙ‫خ ا‬١ٌ‫أعبط اٌّززب‬
Common ratio
ٟ‫ٕذع‬ٌٙ‫عؾ ا‬ٌٛ‫ا‬
Geometric average (mean)
The first term
‫ي‬ٚ‫اٌؾذ األ‬
The last term
‫ش‬١‫اٌؾذ األخ‬
The nth term
ٌٟٕٔٛ‫اٌؾذ ا‬
َ‫اٌؾذ اٌؼب‬
The general term
Consecutive terms
‫خ‬١ٌ‫د ِززب‬ٚ‫ؽذ‬
Respectively
ٌٟ‫ا‬ٛ‫ اٌز‬ٍٝ‫ػ‬
Sets
‫ػخ اٌشبٍِخ‬ّٛ‫اٌّغ‬
Universal set
)
Belonging
 ٖ‫االٔزّبء ( سِض‬
‫ػذَ أزّبء‬
Not belonging
ٞٚ‫خ اٌزغب‬١‫خبط‬
Equality property
‫خ‬٠ٚ‫ػبد ِزغب‬ّٛ‫ِغ‬
Equal sets
‫خ‬١ٌ‫ػخ اٌخب‬ّٛ‫اٌّغ‬
Empty (null) set
ٞٚ‫ اٌزغب‬ٚ‫اح أ‬ٚ‫اٌّغب‬
Equality
‫ػخ‬ّٛ‫ِىٍّخ اٌّغ‬
Complement of a set
‫اٌفشق‬
Difference
ٓ١‫ػز‬ّٛ‫ٓ ِغ‬١‫اٌفشق ث‬
Difference of two sets
‫خ‬١ٙ‫ِٕز‬
Finite
) ‫خ‬١‫بئ‬ٙٔ ‫خ ( ال‬١ٙ‫ش ِٕز‬١‫غ‬
Infinite
‫خ‬١ٙ‫ش ِٕز‬١‫ػخ غ‬ّٛ‫ِغ‬
Infinite set
‫رمبؽغ‬
Intersection
205
ٓ١‫ػز‬ّٛ‫رمبؽغ ِغ‬
Intersection of two sets
‫ارؾبد‬
Union
ٓ١‫ػز‬ّٛ‫ارؾبد ِغ‬
Union of two sets
‫ػخ لبثٍخ ٌٍؼذ‬ّٛ‫ِغ‬
Denumerable set ( countable set )
) ‫ضح‬١ِّ ‫خ ( طفخ‬١‫خبط‬
Characteristic
) ( ْ‫شا‬١‫عبْ طغ‬ٛ‫ل‬
Brackets
Braces
ْ‫شا‬١‫عبْ وج‬ٛ‫ل‬
‫اط‬ٛ‫أل‬
Parentheses
‫خ‬ٙ‫اط ِزشبث‬ٛ‫أل‬
Like brackets
‫ػبد‬ّٛ‫خ اٌّغ‬٠‫ٔظش‬
Set theory
‫خ‬١‫ػخ عضئ‬ّٛ‫ِغ‬
Subset
‫ػخ‬ّٛ‫ ِغ‬ٟ‫ػٕظش ف‬
Element of a set
‫مخ اٌمبئّخ‬٠‫ؽش‬
Listing method
Solid geometry
‫خ‬١‫ٕ٘ذعخ فشاغ‬
Solid geometry
‫فؼبء‬
Space
) ‫فشاؽ ( ٔظف فؼبء‬
Half space
‫اٌىشح‬
Sphere
Center of a sphere
‫ِشوض وشح‬
Chord of a sphere
‫رش اٌىشح‬ٚ
‫الد‬١‫ ِغزط‬ٞ‫اص‬ٛ‫ِز‬
Cuboid
‫ِىؼت‬
Cube
‫أخ‬ٛ‫اعط‬
Cylinder
‫خ اٌمبئّخ‬٠‫أخ اٌذائش‬ٛ‫االعط‬
Right circular cylinder
ٟٔ‫ا‬ٛ‫ً اٌغطؼ األعط‬١ٌ‫د‬
Directrix of a cylindrical
‫ػ‬ٛ‫ش اٌغط‬١‫لطش وض‬
Diagonal of a polyhedron
‫ؽ‬ٚ‫اٌّخش‬
Cone
206
ٟ‫ؽ‬ٚ‫ً اٌمطغ اٌّخش‬١ٌ‫د‬
Directrix of a conic
‫س‬ٛ‫إٌّش‬
Prism
ٟ‫س عذاع‬ٛ‫ِٕش‬
Hexagonal prism
ُ‫س لبئ‬ٛ‫ِٕش‬
Right prism
‫ؽ‬ٚ‫لبػذح اٌّخش‬
Base of a cone
َ‫٘ش‬
Pyramid
ٟ‫٘شَ صالص‬
Tetrahedron
)‫خ األػالع‬٠ٚ‫ٗ ِضٍضبد ِزغب‬ٙ‫ع‬ٚ‫غ أ‬١ّ‫ ِٕزظُ ( ٘شَ ع‬ٟ‫٘شَ صالص‬
Regular tetrahedron
َ‫سأط ٘ش‬
Vertex of pyramid
َ‫ش‬ٌٍٙ ٟ‫عٗ اٌغبٔج‬ٌٛ‫ا‬
Lateral face of a pyramid
‫ػبئٍخ‬
Family
‫بد‬١ٕ‫ػبئٍخ ِٓ إٌّؾ‬
Family of curves
‫ػ‬ٛ‫ػبئٍخ ِٓ اٌغط‬
Family of surfaces
Geometric figure
ٟ‫شىً ٕ٘ذع‬
ٞٛ‫شىً ِغز‬
Plane figure
Capacity
‫اٌغؼخ‬
Volume
ُ‫ؽغ‬
ّٟ‫ؽغ‬
Volumetric
Volume of a solid
ُ‫ؽغُ ِغغ‬
Rectangular solid
ُ‫ِغغُ لبئ‬
Dense
‫ف‬١‫وض‬
Density
‫وضبفخ‬
Depth
‫ػّك‬
‫خ‬١‫اٌّغبؽخ اٌغبٔج‬
Lateral area
‫خ‬١ٍ‫اٌّغبؽخ اٌى‬
Total area
‫خ‬١ٍ‫خ اٌى‬١‫اٌّغبؽخ اٌغطؾ‬
Total surface area
207
Statistics
‫اإلؽظبء‬
Statistics
Statistical data
‫خ‬١‫بٔبد ئؽظبئ‬١‫ث‬
Statistical analysis
ٟ‫ً ئؽظبئ‬١ٍ‫رؾ‬
)‫ش٘ب‬١‫غ‬ٚ ‫عؾ – اٌزشزذ‬ٛ‫ – اٌّز‬ٜ‫غبد اٌّذ‬٠‫بٔبد( ئ‬١‫ ٌٍج‬ٟ‫ً اإلؽظبئ‬١ٍ‫اٌزؾ‬
Statistical analysis of data
‫خ‬٠‫إٌضػخ اٌّشوض‬
Central tendency
‫االعزذالي‬
Inference
ٟ‫االعزذالي اإلؽظبئ‬
Statistical inference
Ordered data
‫بٔبد ِشرجخ‬١‫ث‬
Data representation
‫بٔبد‬١‫ً اٌج‬١‫رّض‬
‫ اٌّؼذي‬ٚ‫عؾ أ‬ٛ‫اٌّز‬
Average
Correlation
‫اسرجبؽ‬
Variance
ٓ٠‫اٌزجب‬
) ‫ االؽزّبالد‬ٟ‫ٓ ( ف‬٠‫غ راد اٌؾذ‬٠‫ص‬ٛ‫ر‬
Binomial distribution
‫ؽذس‬
Event
‫ادس ِغزمٍخ‬ٛ‫ؽ‬
Independent events
‫ش ِغزمٍخ‬١‫ادس غ‬ٛ‫ؽ‬
Dependent events
) ْ‫ب‬١‫ؽبدصبْ ِٕفظالْ ( ِزٕبف‬
Mutually exclusive events
‫اٌؾذس اٌّإوذ‬
Certain (Sure) event
ٟ‫ِؼبًِ االسرجبؽ اٌخط‬
Linear correlation coefficient
) r= ± 1( َ‫اسرجبؽ رب‬
Perfect correlation
‫ٕخ‬١‫ػ‬
Sample
‫ِؼبًِ اسرجبؽ اٌشرت‬
Rank correlation coefficient
‫رشزذ‬
Dispersion
‫ٕخ‬١‫فشاؽ اٌؼ‬
Sample space
‫ارظ‬ٌٕٛ‫فشاؽ ا‬
Outcome
‫ؽغش إٌشد‬
Dice
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ً١‫ؽبدس ِغزؾ‬
Null ( empty ) event
‫لغ‬ٛ‫ر‬
Expectation
ٟ‫ائ‬ٛ‫ػش‬
Random
‫خ‬١‫ائ‬ٛ‫ٕخ ػش‬١‫ػ‬
Random sample
ْٛ‫شع‬١‫ِؼبًِ ث‬
Pearson's coefficient
‫اؽزّبي‬
Probability
ٟ‫ائ‬ٛ‫ش ػش‬١‫ِزغ‬
Random variable
‫اٌّشب٘ذاد‬
Observations
Mode
‫اي‬ٌّٕٛ‫ا‬
Median
‫ؾ‬١‫ع‬ٌٛ‫ا‬
‫رىشاس‬
Frequency
Frequency distribution
ٞ‫غ اٌزىشاس‬٠‫ص‬ٛ‫اٌز‬
Frequency Table
ٞ‫ي اٌزىشاس‬ٚ‫اٌغذ‬
Cumulative frequency
ّٟ‫اٌزىشاس اٌزشاو‬
‫اٌفئبد‬
Sets
Histogram
ٞ‫اٌّذسط اٌزىشاس‬
Frequency polygon
ٞ‫اٌّؼٍغ اٌزىشاس‬
‫اٌّزغّغ اٌظبػذ‬
Ascending cumulative
‫ اٌّزغّغ اٌظبػذ‬ٞ‫ي اٌزىشاس‬ٚ‫اٌغذ‬
Ascending cumulative frequency table
‫اٌّزغّغ إٌبصي‬
Descending cumulative
‫ اٌّزغّغ إٌبصي‬ٞ‫ي اٌزىشاس‬ٚ‫اٌغذ‬
Descending cumulative frequency table
‫ِشوض فئخ‬
Center of class
‫خ‬١‫اؽزّبالد ششؽ‬
Conditional probabilities
Standard deviation
ٞ‫بس‬١‫االٔؾشاف اٌّؼ‬
Average (Mean ) deviation
‫عؾ‬ٛ‫االٔؾشاف اٌّز‬
Quartile deviation
ٟ‫ؼ‬١‫االٔؾشاف اٌشث‬
Coefficient of variation
‫ِؼبًِ االخزالف‬
Coefficient of regression
‫ِؼبًِ االٔؾذاس‬
‫ح‬ٚ‫ػال‬
Bonus
209
ْ‫ئؽظبء اٌغىب‬
Census
)‫د أخطبء‬ٛ‫ع‬ٚ َ‫اخزجبس( ٌٍزأوذ ِٓ ػذ‬
Check
ٟ‫ٕذع‬ٌٙ‫اٌشعُ ا‬
Geometrical drawing
‫ً ثبألػّذح‬١‫اٌزّض‬
Bar graph
‫خ‬٠‫ً ثبٌمطبػبد اٌذائش‬١‫اٌزّض‬
Circular graph (pie graph )
ُ١‫رٕظ‬
Organization
Rate of consumption
‫الن‬ٙ‫ِؼذي االعز‬
Monthly salary
ٞ‫ش‬ٙ‫اٌشارت اٌش‬
‫ٌخ‬ّٛ‫ػ‬
Commission
) ٖ‫بس‬١‫زُ اخز‬٠ ُ‫ػذ( ٔجذأ اٌؼذ ِٓ سل‬
Count
Trigonometry
Trigonometry
‫ؽغبة اٌّضٍضبد‬
Periodic curve
ٞ‫س‬ٚ‫ د‬ٟٕ‫ِٕؾ‬
‫خ االٔخفبع‬٠ٚ‫صا‬
Angle of depression
‫خ االسرفبع‬٠ٚ‫صا‬
Angle of elevation
‫ي‬ٚ‫ اٌشثغ األ‬ٟ‫خ ف‬٠ٚ‫صا‬
First quadrant angle
Negative angle
‫خ عبٌجخ‬٠ٚ‫صا‬
Positive angle
‫عجخ‬ِٛ ‫خ‬٠ٚ‫صا‬
Initial of point
‫خ‬٠‫ٔمطخ اٌجذا‬
Terminal point
‫خ‬٠‫ب‬ٌٕٙ‫ٔمطخ ا‬
‫خ‬ٙ‫خ ِزغ‬٠ٚ‫صا‬
Directed (oriented=sensed) angle
‫خ‬٠‫لطؼخ دائش‬
Segment of a circle ( circular segment)
Major segment
ٞ‫لطؼخ وجش‬
Standard position
ٟ‫بع‬١‫ػغ ل‬ٚ
ٟ‫بع‬١‫ػغ ل‬ٚ ٟ‫خ ف‬٠ٚ‫صا‬
Angle in standard position
Clockwise
‫ارغبٖ ػمبسة اٌغبػخ‬
Counterclockwise (Anti-Clockwise)
‫ػىظ ػمبسة اٌغبػخ‬
210
Compass
‫طٍخ‬ٛ‫ث‬
Compasses
‫فشعبس‬
Cosine ( Cos )
) ‫ت اٌزّبَ ( عزب‬١‫ع‬
Secant ( Sec)
) ‫خ ( لب‬٠ٚ‫لبؽغ اٌضا‬
Sine ( Sin )
) ‫خ ( عب‬٠ٚ‫ت اٌضا‬١‫ع‬
Cosecant (Csc = cosec)
) ‫لبؽغ اٌزّبَ ( لزب‬
Tangent ( Tan)
) ‫خ ( ظب‬٠ٚ‫ظً اٌضا‬
) ‫خ ( ظزب‬٠ٚ‫ظً رّبَ اٌضا‬
Cotangent ( Cot)
‫دسعخ‬
Degree
‫خ‬٠ٚ‫ػؼف اٌضا‬
Double angle
‫لطبع‬
Sector
ٞ‫لطبع دائش‬
Sector of a circle
ٟٕ١‫ش ) اٌغز‬٠‫بط(اٌزمذ‬١‫اٌم‬
Sexagesimal measure
The sixtieth measure
ٟٕ١‫ش اٌغز‬٠‫اٌزمذ‬
Circular measure
ٞ‫بط اٌذائش‬١‫اٌم‬
ٗ١‫ٔغت ِضٍض‬
Trigonometric ratios
‫خ‬١‫اٌّؼبدالد اٌّضٍض‬
Trigonometric equations
Law of sine
‫ت‬١‫ْ اٌغ‬ٛٔ‫لب‬
Initial side
ٟ‫ػٍغ اثزذائ‬
Tangent curve
ً‫ اٌظ‬ٟٕ‫ِٕؾ‬
ً‫ْ اٌظ‬ٛٔ‫لب‬
Tangent law
‫طٍخ‬ٛ‫ث‬
Compass
‫ؽً اٌّضٍش‬
Solution of a triangle
Variables
‫ش‬١‫ِزغ‬
Variable
‫ش ربثغ‬١‫ِزغ‬
Dependent variable
ً‫ِغزم‬
Independent
211
ً‫ش ِغزم‬١‫ِزغ‬
Independent variable
Vectors
ٗ‫ِزغ‬
Vector
Zero vector
ٞ‫اٌّزغٗ اٌظفش‬
Unit vector
‫ؽذح‬ٌٛ‫ِزغٗ ا‬
Position vector
‫ػغ‬ٌّٛ‫ِزغٗ ا‬
Vector addition
‫بد‬ٙ‫عّغ ايِزغ‬
Sum of vectors
‫بد‬ٙ‫ع ِزغ‬ّٛ‫ِغ‬
ٗ‫اٌؼشة اٌّزغ‬
Vector Multiplication
Dot product =inner product=dot
multiplication
ٍٟ‫اٌؼشة اٌذاخ‬
Vector quantity
) ‫ اٌغشػخ‬- ‫ح‬ٛ‫خ( ِضً اٌم‬ٙ‫خ ِزغ‬١ّ‫و‬
Norm of vector
ٗ‫بس اٌّزغ‬١‫ِؼ‬
Free vector
‫ اٌؼبدح ِطٍك ئال ئرا ؽذدد ٔمطخ‬ٟ‫ ف‬ٛ٘ ٗ‫وً ِزغ‬ٚ ‫ش ِؾذدح‬١‫شٖ غ‬١‫ ِزغٗ ٔمطخ رأص‬ٛ٘ (‫ِزغٗ ِطٍك‬
) ٖ‫ش‬١‫رأص‬
)‫خ ِٕزظّخ (صبثزخ‬ٙ‫عشػخ ِزغ‬
Uniform velocity
Vector algebra
‫بد‬ٙ‫عجش اٌّزغ‬
Vector analysis
‫بد‬ٙ‫ً اٌّزغ‬١ٍ‫رؾ‬
Vector geometry
‫بد‬ٙ‫ٕ٘ذعخ اٌّزغ‬
Vector field
ٟٙ‫ِغبي ِزغ‬
Vector operator
ٟٙ‫ػبًِ ِزغ‬
‫ٍِخض‬
Summary
212
213
214