Mathematics A30
Module 2
Lesson 18
Mathematics A30
Angles and Trigonometric Ratios
Part I
445
Lesson 18
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Lesson 18
Angles and Trigonometric Ratios
Introduction
Lesson 18 will expand your knowledge of the six trigonometric ratios for all possible
angles. You will examine what occurs to the ratios as the terminal arm, of an angle in
standard position, revolves through the coordinate system. Reference angles are also
introduced for a quicker method of solving larger angles.
The Primary Ratios are sine, cosine, and tangent.
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Lesson 18
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Lesson 18
Objectives
After completing this lesson, you will be able to
• determine coterminal angles for a given angle.
• determine the value of the six trigonometric ratios when given a point on the
terminal arm of an angle in standard position.
• determine the reference angle for positive or negative angles.
• determine the signs of the trigonometric ratios for any quadrant.
• determine the values for the six trigonometric ratios, when given one
trigonometric ratio and the quadrant in which the angle terminates.
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Lesson 18
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Lesson 18
18.1 Coterminal Angles
Three Diagrams showing positive angles in standard position.
In lesson 17, standard position was defined and it was noted that the terminal arm of an
angle could fall anywhere in the coordinate system. When the terminal arm is rotated
counterclockwise, as in the above three angles, a positive angle results.
An angle is in standard position when the initial arm lies on the positive x-axis with
its vertex at the origin (0,0).
In lesson 18, we will start by looking at negative angles in standard position.
Diagram of three negative angles in standard position.
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Lesson 18
•
Notice the direction of rotation for negative angles is clockwise.
•
A positive or negative angle is determined by the direction of rotation.
Counterclockwise Rotation = Positive Angle
Clockwise Rotation = Negative Angle
Example 1
Draw an angle whose measure is:
i)
ii)
iii)
35 
 35 
 125 
Solution:
Use your protractor to measure the angles.
i)
ii)
iii)
Example 2
In what quadrant does an angle, in standard position lie, if its measure is:
i)
ii)
iii)
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 190 
 295 
 160 
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Lesson 18
Solution:
i)
ii)
Quadrant Two
iii)
Quadrant One
Quadrant Three
Different angles, placed in standard position, may have the same terminal arm.
Example:
Angles in standard position whose terminal arms
coincide are called coterminal angles.
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Lesson 18
Example 1
Determine three positive and three negative coterminal angles for 50 .
Solution:
Positive Angles – Counter clockwise Rotation
y
y
410°
50  360  410
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y
770°
x
50  360  360  770
454
1130°
x
x
50  360  360  360  1130
Lesson 18
Negative Angles - Clockwise Rotation
y
y
x
– 310°
x
x
– 670°
50   360   310
•
y
– 1030°
50    360    360    670 
50   360   360   360  1030
There are infinitely many angles coterminal with a given angle.
The principle angle of any given angle in standard
position is the smallest positive angle in standard
position which is coterminal with the given angle.
Example 2
A)
B)
What is the principal angle whose measure is  240 ?
What is the principal angle whose measure is 840 ?
Solution:
A)
B)
y
120°
x
840°
Principal angle
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120 
Principal angle
455
120 
Lesson 18
•
If the principal angle has a measure of 50  then all other angles coterminal (as seen
in Example 1) with it can be written 50  + n 360 , n  I , where n is any integer.
I = ...  3 ,  2 ,  1, 0 , 1, 2 , 3...
Check
If n = 0,
the measure is 50  .
If n = 1,
If n = 2,
If n = 3,
the measure is 410  .
the measure is 770  .
the measure is 1130  .
If n =  1 ,
If n =  2 ,
If n =  3 ,
the measure is  310  .
the measure is  670  .
the measure is  1030  .
}
}
Principal Angle
Positive
Angles
Negative
Angles
General Form - For writing coterminal angles
If A represents any angle, then all angles coterminal with A are
represented by  + n 360 , n  I , where  is the principal angle.
Example 3
Determine the general form of the coterminal angles for an angle whose
measure is  215  .
Solution:
Find the principal angle.
•
Sketch given angle.
215° = 180° + 35°
•
Locate smallest, positive angle
180   35   145 
The principal angle measures 145°.
Substitute principal angle into general form equation.
145   n 360 , n  I
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Lesson 18
Exercise 18.1
1.
Determine three positive and three negative coterminal angles for each of the
following.
a.
b.
c.
d.
e.
f.
2.
Find the principal angle for each of the following angles.
a.
b.
c.
d.
e.
f.
3.
60°
125°
305°
180°
 75 
 105 
 30 
690°
 330 
 315 
780°
1065°
Write the general form of the coterminal angles for each of the following.
a.
b.
c.
d.
45°
173°
 50 
 243 
18.2
Trigonometric Ratios and Trigonometric
Functions
The relationships between the angles and the sides of a right-angled triangle are called
the trigonometric ratios or trigonometric functions and these ratios are the foundation of
trigonometry. You have seen these ratios in Math 20 and they will now be reviewed.
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Lesson 18
•
In the diagram  is an acute angle in standard position. AB is any perpendicular
drawn to the initial ray OB. Hence,
OA is the hypotenuse,
AB is the side opposite to angle , and
OB is the side adjacent to angle .
We define the trigonometric ratios of angle  with reference to the sides of this right-angle
triangle as follows:
sine 
•
sin  
or
opposite side AB

hypotenuse OA
cosine 
or
cos  
adjacent side OB

hypotenuse
OA
tangent 
or
tan  
opposite side AB

adjacent side OB
cosecant 
or
csc  
hypotenuse OA

opposite side AB
secant 
or
sec  
hypotenuse
OA

adjacent side OB
cotangent 
or
cot  
adjacent side OB

opposite side AB
Sine, cosine and tangent are often referred to as the three primary ratios in
trigonometry.
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Lesson 18
*
An easy way to remember the primary trigonometry ratios is to use the
abbreviation:
SOH CAH TOA
Sin  
•
Opp
Hyp
Cos  
Adj
Hyp
Opp
Adj
Notice that the cosecant, secant, and cotangent ratios are reciprocals of the sine,
cosine, and tangent ratios.
csc  =
1
sin 
sec  =
1
cos 
cot  =
1
tan 
Another relationship worth mentioning is tan  =
sin 
=
cos 
=
=
=
•
Tan  
 AB 


 OA 
 OB 


 OA 
AB OA

OA OB
AB
OB
tan 
Similarly, the relationship cot  =
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sin 
since
cos 
cos 
can be proven.
sin 
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Lesson 18
Problems involving the trigonometric ratios can be solved in two ways. One way is with
the use of a scientific calculator and another is with the use of a table of trigonometric
values.
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Lesson 18
•
A scientific calculator has the SIN, COS and TAN function keys.
•
Example: Find sin 37  .
CLEAR SIN 37 ENTER
Display: 0.6018
*
If you did not get 0.6018 as an answer for sin 37  , then you need to check to
see if your calculator is reading 37 rad (radians) instead of 37  .
MODE
With cursor key arrow down  twice and arrow right  once
to have Degree highlighted.
ENTER
CLEAR
(to get back to the original screen)
Try the above box again!
•
Example: Find csc 37  .
CLEAR SIN 37 ENTER
1
ENTER
x
Display:
1.6616
csc  =
1
sin 
x
1
=
1
x
1
=
1
x
•
There is also a table of trigonometric functions. This table has been inserted into
this lesson and can be used in place of the calculator.
•
Always round your answer to 4 decimal places.
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Lesson 18
Trigonometric Table
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Lesson 18
Example 1
In the right-angled triangle ABC, AB = 3,
BC = 4, and AC = 5. State the six
trigonometric ratios of C and of A.
Solution:
opp 4

hyp 5
adj 3
cos A 

hyp 5
opp 4
tan A 

adj 3
1
3
cot A 

tan A 4
1
5
sec A 

cos A 3
1
5
csc A 

sin A 4
opp 3

hyp 5
adj 4
cos C 

hyp 5
opp 3
tan C 

adj 4
1
4
cot C 

tan C 3
1
5
sec C 

cos C 4
1
5
csc C 

sin C 3
sin C 
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sin A 
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Lesson 18
The definitions of the basic trigonometric ratios in terms of the sides of a right triangle
can be applied to any acute angle in standard position on the coordinate plane.
With a point P(x,y) on the terminal arm, we can find the distance from P(x,y) to the
origin. This was known as r (as in radius) in lesson 17.

•
Start P(x,y) at point S(r,0).
•
As P(x,y) travels about the coordinate
system, a circular path is produced.
•
The angle  is the amount of rotation about the origin.
•
The radius of the circle is the same as the length of the terminal arm (r).
When finding the distance from P(x,y) to the origin, a perpendicular segment is
constructed from P(x,y) to the x-axis so that a right triangle is produced. With the use of
the Pythagorean Theorem, the distance (r) is found. r 2  x 2  y 2
(Remember, length is always positive.)
The sides of the triangle are x
and y in length and the
hypotenuse has length r.
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Lesson 18
The six trigonometric ratios can now be redefined as follows for any angle in the first
quadrant of measure  in standard position and any point (x, y) on the terminal arm.
sin  =
y
opp
=
r
hyp
csc  =
hyp
r
=
opp
y
cos  =
x
adj
=
r
hyp
sec  =
r
hyp
=
x
adj
tan  =
y
opp
=
x
adj
cot  =
adj
x
=
opp
y
Example 2
The terminal ray of an angle  in standard position passes through P(8,15).
Evaluate the six trigonometric ratios for  .
Solution:
Draw a perpendicular from P to the x-axis to form a right triangle.
2
2
2
r = x  y
2
2
2
r = 8   15 
2
r = 64  225
2
r = 289
r = 17 units
sin  =
y
15
=
r
17
csc  =
r
17
2
=
= 1
y
15
15
cos  =
x
8
=
r
17
sec  =
r
17
1
=
= 2
x
8
8
tan  =
y
15
=
x
8
cot  =
x
8
=
y
15
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Lesson 18
Using the values for x, y, and r rather than opp, adj, and hyp, has the advantage that
trigonometric ratios may be found of angles in standard position in any quadrant. The
same definitions of the ratios apply. Try the next example when the terminal arm is in
quadrant two.
Example 3
Evaluate the six trigonometric ratios for angle A whose terminal ray passes through
the point  2, 4  .
Solution:
y
2
2
2
r = x + y
2
2
2
r =  2  + 4 
2
r = 4+ 16
P (–2, 4)
2
r = 20
r
4
r =
A
–2
x
20 = 2 5
x = 2
y= 4
r= 2 5
Note that, although r must always be positive, x and y may be positive or negative
depending on the coordinates of P.
sin A =
=
=
y
4
=
r
2 5
2
csc A =
=
5
5
5
=
x
2
=
r
2 5
1
=
5
sec A =
cos A =
= 
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=
=
5
5
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r
y
2 5
4
5
2
r
x
2 5
2
 5
Lesson 18
x
y
2
=
4
1
= 
2
y
tan A =
x
4
=
2
= 2
cot A =
If an angle in standard position has its terminal ray coinciding with one
of the x or y axis, it is called a quadrantal angle.
Example 4
Evaluate the six trigonometric functions of the quadrantal angle whose measure is
180°.
Solution:
Let P(x,y) be any point on the terminal side of the angle. In particular, pick P =  2,0  .
2
2
2
r = x + y
2
2
r =  2  + 0 
2
r = 4
r = 2
x = 2
y= 0
r= 2
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Lesson 18
r
2
=
y
0
= Undefined
y
0
=
r
2
= 0
csc 180  =
sin 180  =
x
2
=
r
2
= 1
r
2
=
x
2
= 1
x
2
cot 180  =
=
y
0
= Undefined
cos 180  =
sec 180  =
y
0
=
x
2
= 0
tan 180  =
Note: Any other point P(x,y) would have given the same answers.
You may repeat the example using P  6,0  .
Division by zero is undefined.
Example 5
Evaluate sin, cos, and tan of the quadrantal angle 270°.
Solution:
Let P(x,y) be any point on the terminal side of the angle.
In particular, let the point be P 0 ,  3  .
2
2
2
r = x +y
2
2
2
r = 0  +  3 
2
r = 9
r = 3
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Lesson 18
y
3
=
r
3
= 1
x= 0
sin 270  =
y = 3
r= 3
x
0
=
r
3
= 0
y
3
tan 270  =
=
x
0
= Undefined
cos 270  =
Exercise 18.2
1.
Complete the following table. Use the three diagrams found on the next page
to state the six trigonometric ratios of the acute angles in the right-angled
triangles.

22.6°
sin 
opp 5
=
hyp 13
67.4°
30°
60°
45°
cos 
tan 
csc 
sec 
cot 
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Lesson 18
67.4°
13
5
22
1
22.6°
12
45°
1
2
1
30°
33
2.
Prove the relationship cot  =
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cos 
using the diagram.
sin 
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Lesson 18
3.
Determine the six trigonometric ratios for each of the following ordered pairs, where
the point is on the terminal arm of an angle in standard position.
(Do not express as decimals.)
a)
 5 ,  12 
b)
2 ,  3 
c)
 5, 0 
d)
(6, 1)
e)
(0, 7)
f)
 40 , 9 
18.3 Reference Angles
Any angle A in standard position has an associated acute angle called a reference angle.
Reference angles are useful for converting trigonometric functions of any large angle into
a trigonometric function of an angle between 0° and 90°.
The reference angle of a given angle A in standard position
is the positive acute angle determined by the x-axis and
the terminal side of the given angle.
•
In the following diagrams,  denotes the reference angle of angle A in each of the
four quadrants. ( m A is read 'measure of angle A'.)
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Lesson 18
In each of the diagrams,  is the positive angle between the terminal arm and the nearest
position of the x-axis.  is always positive and is the reference angle.
QUADRANT TWO
QUADRANT ONE
m  180   mA
m   m A
QUADRANT THREE
QUADRANT FOUR
m  mA  180 
m  360   mA
Steps to Calculate the Reference Angle
•
Make a sketch of the given angle. The sketch will show if the terminal side of the
given angle is closer to the positive x-axis or to the negative x- axis.
•
Calculate the measure of the angle between the terminal ray and the nearest xaxis. The reference angle is always positive and no greater than 90°.
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Lesson 18
Example 1
Sketch each angle whose measure is given and determine the measure of the
reference angle.
a)
b)
c)
d)
m A =
m A =
m A =
m A =
 70 
150 
 210 
800 
Solution:
a)
Make sketch of the given angle.
Calculate the measure of the angle between
the terminal ray and the nearest x-axis.
b)
Reference angle has
measure 70°.
Make sketch of the given angle.
Calculate the measure of the angle between
the terminal ray and the nearest x-axis.
has
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180  150  30
Reference angle
Lesson 18
measure 30°.
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Lesson 18
c)
Make sketch of the given angle.
Calculate the measure of the angle between
the terminal ray and the nearest x-axis.
has
d)
measure 30°.
Make sketch of the given angle.
Calculate the measure of the angle between
the terminal ray and the nearest x-axis.
has
•
210  180  30
Reference angle
800° = (2 × 360°) + 80 °
Reference angle
measure 80°.
All coterminal angles have the same reference angles.
Example 2
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Lesson 18
Find the reference angle of an angle whose measure is 200° and of an angle whose
measure is  160  .
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Lesson 18
Solution:
Sketch the given angle.
Calculate the measure of the angle between
the terminal ray and the nearest x-axis.
200  180  20
180  160  20
Reference angle has measure 20°.
In general form, any angle with measure
200 + n360, n  I , has a reference angle of 20°.
Why are reference angles important?
Before this question can be answered, we need to study the outcome of the signs for the
trigonometric ratios depending on which quadrant the terminal arm lies. Since the
coordinates x and y vary in sign from quadrant to quadrant, the trigonometric function
varies in sign from quadrant to quadrant.
The Sign of a Trigonometric Ratio
•
Do the following activity to refresh your memory as to which trigonometric function
is positive or negative for an angle whose terminal arm lies in a specific quadrant.
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Lesson 18
Activity 18.3
• Determine the six trigonometric ratios for each point on the terminal arm of the
angle. Fill in the chart.
a)
b)
r 2  x2  y2
sin  
y
r
cos  
x
r
tan  
y
x
csc  
r
y
sec  
r
x
cot  
x
y
(3, 4)
 3, 4 
c)
d)
 3,  4 
3,  4 
Quadrant
One
(3, 4)
Quadrant
Two
 3, 4 
Quadrant
Three
 3,  4 
Quadrant
Four
3,  4 
x= 3
x = 3
x = 3
x= 3
y= 4
r= 5
y= 4
r= 5
y= 4
r= 5
y= 4
r= 5
4
5

3
5
4
1
= 1
3
3
Reference
Angle
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Lesson 18
•
Notice that all six trigonometric functions have positive values for angles in the
first quadrant because x, y, and r are each positive.
•
In the second quadrant sine (and its reciprocal ratio, cosecant) is positive but cosine
and tangent (and their reciprocal ratios) have negative values.
•
In the third quadrant tangent (and its reciprocal ratio, cotangent) is positive but
sine and cosine (and their reciprocal ratios) have negative values.
•
In the fourth quadrant cosine (and its reciprocal ratio, secant) is positive but sine
and tangent (and their reciprocal ratios) have negative values.
The above four statements can be summarized into a simple saying that will help you
remember the signs of the trigonometric ratios for any quadrant.
The CAST Rule.
This rule identifies the trigonometric ratios (and the reciprocals)
that are positive in each quadrant.
C - Cosine
A - All
S - Sine
T - Tangent
Example 3
If A is an angle whose sine is positive, sketch two possible angles in the correct
quadrants.
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Lesson 18
Solution:
Since sine is positive in the first and second quadrants the two possible angles are shown
in the diagram. The measure of angle A is not known. The main concern is to draw a
typical angle in the correct quadrant.
Example 4
Sketch the angle A in the correct quadrant which satisfies simultaneously the
conditions that csc A is negative and cot A is positive.
Solution:
For csc A < 0,
angle A is in
3rd or 4th
quadrant.
For cot A >
0, angle A is
in 1st or 3rd
quadrant.
For ccs A < 0, and cot A
> 0 together, angle A is
in 3rd quadrant only.
Therefore, an angle in the third quadrant satisfies the conditions.
We are now ready to answer the question that was asked earlier.
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Lesson 18
Why are reference angles important?
If you are using the trigonometric tables to evaluate, you will notice that the tables do not
list angles greater than 90°. The reason is that the trigonometric function of any angle
greater than 90°may always be expressed in terms of the same trigonometric function of
the reference angle. (You have seen this in Activity 18.3) Reference angles are always
between 0° and 90°.
Example 5
Express cos 145° in terms of the reference angle and then evaluate.
Solution:
Minus signs are used in labelling the sides to remind us that coordinates are
negative.
Sketch the 145° angle and from any point P on the terminal ray
draw a perpendicular to the x-axis.
The reference angle has a measure of 35°.
(By definition of cosine)
x
r
x
  
r
 cos 35 
 0 .8192
cos 145 
adj
hyp
x
cos 35  
r
cos 35  
use your
calculator
or trig. tables
Mathematics A30
481
Lesson 18
 cos 145    cos 35    0.8192
Check with your calculator
cos 145 
 cos 35 
cos is negative
in Quad. 2
reference angle
Example 5 shows us …
The value of a trigonometric function of angle A is equal to the value of
•
the same trigonometric function of the reference angle of A with the
appropriate + or – sign (depending on the quadrant of the terminal
side of angle A).
The following example is similar to example 5 except that the answer is obtained quicker
when the above principal is applied.
Example 6
i)
ii)
Express cos 145° as a function of the reference angle and evaluate.
Express tan ( 250) as a function of the reference angle and evaluate.
Solution:
i)
The angle is in the second quadrant and the reference angle is 35°.
cos 145  cos 35 
 0 .8192
ii)
cosine is
negative in
the second
quadrant
The terminal arm of the angle is in the 2 quadrant where tan is negative. The
reference angle is 70°.
Mathematics A30
482
Lesson 18
tan ( 250 )   tan 70   2 .7475
Mathematics A30
483
Lesson 18
Exercise 18.3
1.
Draw each angle in standard position having the given measure. Label and give the
measure of the reference angle.
a)
b)
c)
d)
e)
f)
135 
300 
280
172 
 115 
 35 
440 
590 
1200 
1150 
 850 
 865 
g)
h)
i)
j)
k)
l)
2.
If A is an angle whose cosine is positive, sketch two possible angles in the correct
quadrants.
3.
Sketch the angle A in the correct quadrant which simultaneously satisfies the
conditions that cot A is negative and sec A is positive.
4.
Determine the six trigonometric values of each of the following, using reference
angles in each case.
a.
b.
c.
255°
305°
168°
18.4 Finding Trigonometric Ratios When the
Angle is not Given.
Example 1
4
and  is an angle in the third quadrant, sketch and label the angle
5
and find the value of the other five trigonometric ratios.
If cos  = 
Mathematics A30
484
Lesson 18
Solution:
Cosine is negative for angles in the second and third quadrants.
•
The question states that  is in the third quadrant.
Since cos  =
x
4
x
4
=  and since r is always positive, it must be that

or
r
5
r
5
x  4
r 5
•
To complete the labelling of the diagram we must find y.
r2 = x 2 + y 2
5 2
=  4 2 + y 2
y 2 = 25  16
y2 = 9
y = 3
y =  3 is chosen since the y-value
is negative in the third quadrant.
We can conclude,
y
3
=
r
5
x
4
cos  =
= 
r
5
y
3
3
tan  =
=

x
4
4
r
5
2
=
= 1
y
3
3
r
5
1
sec  =
=
= 1
x
4
4
x
4
1
cot  =

 1
y
3
3
sin  =
Mathematics A30
csc  =
485
Lesson 18
Example 2
Evaluate the five other trigonometric functions of  if sin  = 
3
.
5
Solution:
Draw a diagram.
•
Since the quadrant is not specified, there are two possibilities for  because sine is
negative in the third quadrant and fourth quadrant.
Label the triangles.
sin  =
y
3 3
=  
r
5
5
y = 3
r= 5
Calculate the unknown sides and determine the coordinates of P and Q.
2
2
2
r = x +y
52
= x 2 +  3 2
2
x = 25  9
2
x = 16
x = 4
 P =  4,  3 and Q = 4 ,  3  .
Mathematics A30
486
Lesson 18
Evaluate the functions.
Third Quadrant
Fourth Quadrant
sin  =
y
3
=
r
5
sin  =
y
3
=
r
5
cos  =
x
4
=
r
5
cos  =
x
4
=
r
5
tan  =
y
3 3
=

x
4 4
tan  =
y
3
=
x
4
csc  =
r
5
= 
y
3
csc  =
r
5
= 
y
3
sec  =
r
5
= 
x
4
sec  =
r
5
=
x
4
cot  =
x
4
=
y
3
cot  =
x
4
=
y
3
Mathematics A30
487
Lesson 18
Exercise 18.4
1.
2.
Find the exact values in each case. Do not use tables or your calculator to simplify.
a)
Tan  = 5 and  is not in the first quadrant. Find sin  and cos .
b)
Cos  = 
3
and  is in the third quadrant. Find cot  and sin .
4
c)
Sin  = 
1
. Find cos , tan , and csc .
2
d)
Csc  =
5
. Find tan  and cos .
3
Determine the values of all six trigonometric ratios, given the following
information. Leave answer in exact form.
Exact form means no decimals answers.
a)
cos  = 
b)
tan  =
c)
csc  = 
Mathematics A30
2
,  in quadrant two.
3
5
,  in quadrant three.
6
13
,  in quadrant three.
5
488
Lesson 18
Answers to Exercises
Exercise 18.1
1.
a.
 = 60°
60° + 360° = 420°
60° + 360° + 360° = 780°
60° + 360° + 360° + 360° = 1140°
60  360  300
60  360  360  660
60  360  360  360  1020
b.
485°, 845°, 1205°,  235  ,  595,  955
c.
665°, 1025°, 1385°,  55,  415,  775
d.
540°, 900°, 1260°,  180,  540,  900
e.
360  75   285  is the principal angle.
285  360  645
285  360  360  1005
 75  360  435
 435  360  795
 435  360  360  1155
2.
Mathematics A30
f.
255, 615, 975,  465,  825,  1185
a.
Principal angle
489
330°
Lesson 18
b.
330°
c.
30°
d.
45°
e.
360° once around
+ 360°
720° twice around
780°
– 720°
60° left to rotate
Principal angle
3.
f.
345°
a.
45  n360, n  I
b.
173  n360, n  I
c.
310  n360, n  I
Principal angle
d.
Mathematics A30
60°
310°
117  n360, n  I
490
Lesson 18
Exercise 18.2

sin 
cos 
tan 
csc 
sec 
cot 
1.
22.6 
67.4 
30 
opp
5
=
hyp
13
12
13
12
13
adj
5
=
hyp
13
3
2
5
12
12
2
= 2
5
5
1
13
3
= 2
5
5
13
1
=1
12
12
13
1
=1
12
12
13
3
= 2
5
5
2
12
5
5
12
3
=
1
2.
Mathematics A30
1
2
60 
3
2
3
3
3
2
= 2
1
3
1
2
1
2
=
=
1
2
3
=
1
2
3
2 3
3
3
45 

3
=
2
2
=
3
1
=1
1
2 3
3
2
=
1
2
= 2
1
1
=
2
2
2
2
 2
1
3
3
1
=1
1
Follow the same procedure found on page 459.
491
Lesson 18
3.
a)
2
2
2
r = x + y
2
2
2
r =  5  +  12 
2
r = 25 + 144
x  5
y   12
2
r = 169
r = 13
r  13
sin  =
y
 12
=
r
13
csc  
cos  =
x
5
=
r
13
sec  
r
13
=
y
 12
1
 1
12
r 13

x 5
3
 2
5
x
5

y  12
5

12
y
 12
=
x
5
2
= 2
5
cot  
tan  =
b)
2
2
2
r = x +y
2
2
2
r = 2  +  3 
2
r =4+9
2
r = 13
r = 13
x= 2
y = 3
r=
Mathematics A30
492
13
Lesson 18
sin  =
=
cos  =
=
y
3
=
r
13
x
=
r
2
13
sec  =
r
=
x
13
2
2 13
13
y
3
=
x
2
1
= 1
2
d)
r
13
=
y
3
 3 13
13
tan  =
c)
csc  =
cot  =
x
2
=
y
3
sin  =
y
= 0
r
csc  =
r
= Undefined
y
cos  =
x
= 1
r
sec  =
r
= 1
x
tan  =
y
= 0
x
cot  =
x
= Undefined
y
csc  =
r
=
y
=
37
sec  =
r
=
x
y
sin  =
=
r

cos  =
1
37
37
1
37
37
x
=
r
6
37
37
6
6 37
37
y
1
tan  =
=
x
6
=
Mathematics A30
493
Lesson 18
e)
f)
r
1
y
y
7
=
r
7
1
csc  
cos  =
x
= 0
r
sec  =
r
= Undefined
x
tan  =
y
= Undefined
x
cot  =
x
= 0
y
sin  =
y
9
=
r
41
csc  =
cos  =
x
 40
=
r
41
sec  
tan  =
y
9
=
x
 40
cot  =
sin  =
Exercise 18.3
1.
a)
b)
c)
r
41
=
y
9
5
= 4
9
r
41

x  40
1
 1
40
x
 40
=
y
9
4
 4
9
45 
60 
Reference Angle = 80 
Mathematics A30
494
Lesson 18
d)
8
e)
65 
f)
35 
g)
80 
h)
50 
1200  3  360  120
 1080  120
 1080  90  30
i)
60°
1200°
Reference Angle = 60 
j)
70 
 850   2  360    130 
 720    130 
 720    90    40 
k)
50°
l)
Mathematics A30
–850°
Reference Angle = 50°
35 
495
Lesson 18
2.
3.
cot A is negative
4.
a.
sec A is positive
Quadrant Four
sin 255   sin 75  0.9659
cos 255   cos 75  0.2588
tan 255  tan 75  3.7230
csc 255   csc 75
 1.0353
sec 255   sec 75  3.8637
cot 255  cot 75  0.2680
Mathematics A30
496
Lesson 18
b.
sin 305   sin 55  0.8192
cos 305  cos 55  0.5736
tan 305   tan 55  1.4281
csc 305   csc 55  1.2208
sec 305  sec 55  1.7434
cot 305   cot 55  0.7002
ref. angle = 55°
Quad. IV
*only cosine and secant
are positive
c.
sin 168  sin12  0.2709
cos 168   cos 12   0.9781
tan 168   tan 12   0.2126
csc 168  csc 12  4.8097
sec 168   sec 12   1.0223
cot 168   cos 12   4.7046
Mathematics A30
497
Lesson 18
Exercise 18.4
1.
a.
Write tan   5 
5 5

1 1
5
sin  
26
1
cos  
26

 5 26
26

 26
26
b.
cot  
sin  
3
 7

3 7
7
 7
4
c.
cos   
tan   
3
2
1
3

3
3
csc   2
Mathematics A30
498
Lesson 18
d.
3
4
4
cos   
5
tan   
2.
a.
x 2

r
3
2
2
r  x  y2
cos  
3 2   2 2   y 2 
y2  9  4
y 5
x  2
y 5
r 3
y
5

r
3
x 2
cos   
r
3
y
5
tan   
x 2
r
3
3 5
csc   

y
5
5
r
3
1
sec   
 1
x 2
2
x 2 2 5
cot   

y
5
5
sin  
Mathematics A30
499
Lesson 18
b.
sin  
5
61
6

 5 61
61
 6 61
61
61
5 5
tan  

6 6
61
csc  
5
61
sec  
6
6
1
cot  
1
5
5
cos  
c.
Mathematics A30

5
13
 12
cos  
13
5
5
tan  

 12 12
13
3
csc  
 2
5
5
13
1
sec  
 1
 12
12
 12
2
cot  
2
5
5
sin  
500
Lesson 18
Mathematics A30
Module 2
Assignment 18
Mathematics A30
501
Lesson 18
5. Staple the completed
barcode sheet on top of this
address sheet (upper left
corner.)
4. Staple this sheet to the
appropriately-numbered
assignment. Use one
address sheet for each
assignment.
3. Complete the details
in this address box.
2. Number all the pages and
place them in order.
1. Write your name and address
and the course name and
assignment number in the
upper right corner of the first
page of each assignment.
Before submitting your
assignment, please complete
the following procedures:
Postal Code:
City/Town, Province
Street Address or P.O. Box
Name
Country
Print your name and address, with postal code. This address sheet
will be used when mailing back your corrected assignment.
Assignment Number
18
Mark Assigned:
Distance-Learning Teacher’s Name
Course Title
Mathematics A30
Course Number
8404
Student Number
Staple here to
the upper left
corner of your
assignment
Assignment 18
Values
(40)
A.
Multiple Choice: Select the correct answer for each of the following and place
a check () beside it.
1.
The one figure which shows an angle not in standard position is ***.
____
____
____
____
2.
1
2
3
4
The terminal arm of a negative angle in standard position has a
measure of  290 lies in quadrant ***.
____
____
____
____
Mathematics A30
a.
b.
c.
d.
a.
b.
c.
d.
1
2
3
4
501
Lesson 18
3.
The measure of the angle in standard position coterminal with an
angle whose measure is  200 , is ***.
____
____
____
____
4.
a.
b.
c.
d.
30°, 210°
30° + n180°, n  I
30° + n360°, n  I
210° + n360°, n  I
a.
b.
c.
d.
37°
217°
127°
397°
The simplfied form of
____
____
____
____
Mathematics A30
400° + 360°
40°
40° + n360°, nI
40° + n180°, nI
If an angle of measure 757° is one of the angles described by
  n360, n  I , then  is ***.
____
____
____
____
7.
a.
b.
c.
d.
The measures of all the angles in standard position
which are coterminal with either one of the angles
shown in the diagram are given by the expression.
____
____
____
____
6.
160
20°
560°
520°
The measures of all the angles in standard position which are
coterminal with an angle whose measure is 400° are given by the
expression ***.
____
____
____
____
5.
a.
b.
c.
d.
a.
b.
c.
d.
csc 
is ***.
sec 
1
sin cos
tan 
cot 
502
Lesson 18
8.
9.
The simplified form of
____
____
a.
b.
____
c.
____
d.
____
____
____
Mathematics A30
a.
b.
c.
d.
0.0975
0.0974
0.0973
0.0972
a.
b.
c.
d.
10°
11°
79°
80°
Sin A has the same value as ***.
____
12.
cot 2
sin cos 2
By using the trigonometric table of values, if sec  = 5.7587, then  is
***.
____
____
____
____
11.
1
tan 2
If csc  is 10.2593, then sin  is ***.
____
____
____
____
10.
tan 
is ***.
cot 
a.
b.
c.
d.
sin B
cot C
cos B
cos A
The value of tan A is ***.
____
a.
____
b.
____
c.
____
d.
55
3
3
55
73
3
3
33
503
Lesson 18
13.
14.
15.
If B is an angle in standard position and (3, 5) are the coordinates of a
point on the terminal arm of B, then cos B is ***.
____
a.
____
b.
____
c.
____
d.
____
a.
____
b.
____
____
c.
d.
34
5
6
1
2

1
3
1
3
3
3
If the terminal arm of an angle B in standard position coincides with
the negative y-axis then sin B is ***.
a.
b.
c.
d.
0
1
1
undefined
The reference angle of an angle in standard position whose measure is
905° is ***.
____
____
____
____
Mathematics A30
34
3
If  1, 3 are the coordinates of a point on the terminal arm of angle B
in standard position, then tan B is ***.
____
____
____
____
16.
5
a.
b.
c.
d.
180°
5°
 5
185°
504
Lesson 18
17.
The measures of a pair of angles whose reference
angles are the same as the reference angle of B
is ***.
____
____
____
____
18.
280°
100°
80°
10°
a.
b.
c.
d.
1 or 3
2 or 4
3 or 4
4 or 1
If sin   0 and cos   0 , then  is an angle whose terminal arm is in
quadrant ***.
____
____
____
____
Mathematics A30
a.
b.
c.
d.
If tan   0 , then  is an angle whose terminal arm is in quadrant(s)
***.
____
____
____
____
20.
150°,  330
 60 , 60°
150°, 120
120°, 150°
The reference angle of all the coterminal angles given by
280° + n360°, n  I is ***.
____
____
____
____
19.
a.
b.
c.
d.
a.
b.
c.
d.
1
2
3
4
505
Lesson 18
Mathematics A30
506
Lesson 18
Answer Part B and Part C in the space provided. Evaluation of your solution
to each problem will be based on the following.
B.
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
1.
Express each of the following as a trigonometric function of the
reference angle, with the appropriate plus or minus sign and then give
the value to 4 decimal places.
(2)
a.
cos 67
(2)
b.
sin 115
(2)
c.
tan 160
(2)
d.
cos 190°
(2)
e.
cot135
Mathematics A30
507
Lesson 18
(2)
f.
sec 260
(2)
g.
csc 200°
(2)
h.
sin 271°
(2)
i.
csc 175°
(2)
j.
sin 269
2.
(3)
Mathematics A30
Determine the exact values of the required trigonometric ratios.
a.
Find sin A and cos A if tan A 
508
5
and cos A < 0.
3
Lesson 18
1
and tan B > 0.
4
(3)
b.
Find sin B and cot B if cos B  
(3)
c.
Find sec C and sin C if tan C  1 and cos C > 0.
(3)
d.
Find cos D and tan D if sin D 
(3)
e.
Find sin E and csc E if tan E = 5.
Mathematics A30
509
3
.
2
Lesson 18
(5)
3.
On the grids provided draw all possible principal angles in standard
position based on the information given.
a.
b.
c.
d.
e.
a.
1
2
4
cos B 
5
tan A 
9
15
15
sec D  
12
sin E  1
sin C  
y
x
Mathematics A30
510
Lesson 18
b.
c.
y
y
x
d.
e.
y
y
x
Mathematics A30
x
511
x
Lesson 18
(10)
C.
1.
a.
Since the measures of the angles of any
triangle add up to 180°, the measure of B , in
terms of ,in the right triangle ABC is
_______________.
B
A
b.
C
In the first column of blanks, enter the definition of the
trigonometric ratio in terms of the sides of the triangle, using A,
B, C.
In the second column of blanks, enter the appropriate angle
using  .
sin  = _______________ = cos _______________,
cos  = _______________ = sin _______________,
tan  = _______________ = cot _______________,
cot  = _______________ = tan _______________,
sec  = _______________ = csc _______________,
csc  = _______________ = sec _______________.
c.
Mathematics A30
Write a general statement about your observations in part b.
512
Lesson 18
(10)
2.
d.
Refer to the table of trigonometric values in the lesson and show
a test of your statement in c.
e.
If the observation in c is true, explain why trigonometric tables
may be reduced in size to angles from 0° to 45° only.
(STUDENT JOURNAL)
Write a summary of the material in this lesson which is no longer than
one page.
100 
Mathematics A30
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Lesson 18
Mathematics A30
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Lesson 18