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Appendix D.1
D
■
Review of Algebra, Geometry, and Trigonometry
Properties and Measurement
D.1 Review of Algebra, Geometry, and
Trigonometry
■ Algebra
■ Properties of Logarithms
■ Geometry
■ Plane Analytic Geometry
■ Solid Analytic Geometry
■ Trigonometry
■ Library of Functions
Algebra
Operations with Exponents
1. x nx m x nm
冢xy冣
xn
yn
7. cx n c共x n兲
4.
n
2.
xn
x nm
xm
3. 共xy兲 n x ny n
5. 共x n兲 m x nm
8. x n x共n
m
6. x n 共x n兲
兲
m
Exponents and Radicals (n and m are positive integers)
1. x n x x x . . . x
2. x 0 1, x 0
n factors
1
3. xn n , x 0
x
n
5. x1兾n 冪
x
n x a
*4. 冪
x an
n
6. x m兾n 共x1兾n兲m 共冪
x兲
m兾n
m 1兾n
n
x
共x 兲 冪x m
m
2 x 冪x
7. 冪
Operations with Fractions
bc
ad bc
ad
冢 冣 dc 冢bb冣 bd
bd
bd
ad
bc
ad bc
冢 冣 dc 冢bb冣 bd
bd
bd
c
a d
a
b d b d
a
c
a d
2. b d b d
ac
a c
3.
b d
bd
a兾b
a d
4.
c兾d
b c
a兾b a兾b
c
c兾1
1.
冢 冣冢 冣
冢 冣冢 冣 adbc
冢ab冣冢1c 冣 bca
5.
ab b
ac
c
ab ac a共b c兲 b c
ad
ad
d
* If n is even, the principal nth root is defined to be positive.
D1
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Properties and Measurement
Quadratic Formula
ax 2 bx c 0
x
b ± 冪b2 4ac
2a
Factors and Special Products
1. x 2 a2 共x a兲共x a兲
2. x3 a3 共x a兲共x 2 ax a2兲
3. x3 a3 共x a兲共x 2 ax a2兲
4. x 4 a 4 共x a兲共x a兲共x 2 a2兲
Factoring by Grouping
acx3 adx 2 bcx bd ax 2共cx d兲 b共cx d兲 共ax2 b兲共cx d兲
Binomial Theorem
1. 共x a兲2 x 2 2ax a2
2. 共x a兲2 x 2 2ax a2
3. 共x a兲3 x3 3ax 2 3a2x a3
4. 共x a兲3 x3 3ax 2 3a2x a3
5. 共x a兲4 x 4 4ax3 6a2x 2 4a3x a4
6. 共x a兲4 x 4 4ax3 6a2x 2 4a3x a4
n共n 1兲 2 n2 n共n 1兲共n 2兲 3 n3 . . .
ax
ax
nan1x an
2!
3!
n共n 1兲 2 n2 n共n 1兲共n 2兲 3 n3 . . .
ax
ax
± nan1x an
2!
3!
7. 共x a兲n x n nax n1 8. 共x a兲n x n nax n1
Miscellaneous
1. If ab 0, then a 0 or b 0.
2. If ac bc and c 0, then a b.
3. Factorial: 0! 1, 1! 1, 2! 2 1, 3! 3 2
1, 4! 4 3 2 1, etc.
Sequences
1. Arithmetic: a, a b, a 2b, a 3b, a 4b, a 5b, . . .
2. Geometric: ar 0, ar1, ar 2, ar 3, ar 4, ar 5, . . .
a共1 r n1兲
ar 0 ar1 ar 2 ar 3 . . . ar n 1r
1
1
1
1
1
1
3. General harmonic: ,
,
,
,
,
,. . .
a a b a 2b a 3b a 4b a 5b
1 1 1 1 1
4. Harmonic: , , , , , . . .
1 2 3 4 5
1 1 1 1 1
5. p-Sequence: p , p , p , p , p , . . .
1 2 3 4 5
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Appendix D.1
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Review of Algebra, Geometry, and Trigonometry
D3
Series
1
1 共x 1兲 共x 1兲2 共x 1兲3 共x 1兲4 . . . 共1兲n共x 1兲n . . . ,
x
0 < x < 2
1
1 x x 2 x3 x 4 x 5 . . . 共1兲nx n . . . , 1 < x < 1
1x
共x 1兲2 共x 1兲3 共x 1兲4 . . . 共1兲n1共x 1兲n . . .
ln x 共x 1兲 , 0 < x 2
2
3
4
n
x2
x3
x4
x5
xn
ex 1 x . . . . . . , < x < 2! 3!
4!
5!
n!
3
5
7
x
x
x
sin x x . . . , < x < 3! 5! 7!
x2 x 4 x6 . . .
cos x 1 , < x < 2!
4!
6!
k共k 1兲x 2 k共k 1兲共k 2兲x3 k共k 1兲共k 2兲共k 3兲x 4 . . .
共1 x兲k 1 kx , 1 < x < 1*
2!
3!
4!
k共k 1兲x 2 k共k 1兲共k 2兲x3 k共k 1兲共k 2兲共k 3兲x 4 . . .
共1 x兲k 1 kx , 1 < x < 1*
2!
3!
4!
Properties of Logarithms
Inverse Properties
1. ln e x x
2. eln x x
Properties of Logarithms
1. ln 1 0
2. ln e 1
3. ln xy ln x ln y
4. ln
x
ln x ln y
y
5. ln x y y ln x
6. loga x ln x
ln a
*The convergence at x ± 1 depends on the value of k.
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Properties and Measurement
Geometry
Triangles
1. General triangle
Sum of angles
Area β
180
h
θ
α
b
1
1
(base)(height) bh
2
2
2. Similar triangles
β
a
a A
b B
β
A
α
α
b
B
3. Right triangle
c2 a2 b2 (Pythagorean Theorem)
Sum of acute angles 90
α
c
a
β
b
4. Equilateral triangle
Height h Area 60°
冪3s
2
60°
s
s
h
60°
s
冪3s2
4
5. Isosceles right triangle
Area s2
2
45°
s
45°
s
Quadrilaterals (Four-Sided Figures)
1. Rectangle
2. Square
Area 共length兲共width兲 lw
Area 共side兲2 s2
w
s
s
3. Parallelogram
Area bh
4. Trapezoid
1
Area h共a b兲
2
a
h
h
b
b
a
b
h
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Appendix D.1
D5
Review of Algebra, Geometry, and Trigonometry
■
Circles and Ellipses
2. Sector of circle 共 in radians兲
1. Circle
Area r 2
Area Circumference 2 r
sr
r2
2
s
r
θ
r
3. Circular ring
4. Ellipse
Area 共R 2 r 2兲
Area ab
Circumference ⬇ 2
r
R
b
a
Solid Figures
1. Cone 共A area of base兲
Volume h
Ah
3
A
2. Right circular cone
r 2h
3
Lateral surface area r冪r 2 h2
h
Volume r
3. Frustum of right circular cone
共
r
rR 兲h
3
Lateral surface area s共R r兲
Volume r2
R2
R
4. Right circular cylinder
Volume r 2h
s
h
r
h
Lateral surface area 2rh
5. Sphere
4
Volume r 3
3
Surface area 4 r 2
r
冪a
2
b2
2
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Properties and Measurement
Plane Analytic Geometry
Distance Between 冇x1, y1冈 and 冇x2, y2冈
d 冪共x2 x1兲2 共 y2 y1兲2
Midpoint Between 冇x1, y1冈 and 冇x2, y2冈
Midpoint 冢
x1 x2 y1 y2
,
2
2
冣
Slope of Line Passing Through 冇x1, y1冈 and 冇x2, y2冈
y2 y1
x2 x1
m
Slopes of Parallel Lines
m1 m2
Slopes of Perpendicular Lines
m1 1
m2
Equations of Lines
Point-slope form: y y1 m共x x1兲
Vertical line: x a
Slope-intercept form: y mx b
General form: Ax By C 0
Horizontal line: y b
Equations of Circles 关Center: 冇h, k冈, Radius: r兴
Standard form: 共x h兲2 共 y k兲2 r 2
General form: Ax 2 Ay 2 Dx Ey F 0
Equations of Parabolas 关Vertex: 冇h, k冈兴
Axis:
x=h
Axis
Directrix
Vertex
Focus:
(h, k + p)
p<0
Focus
p>0
Vertex:
(h, k)
Directrix:
y=k−p
共x h兲2 4p共 y k兲
(a) Vertical axis: p > 0
(b) Vertical axis: p < 0
Directrix
Directrix:
x=h−p
Vertex
Focus
Axis
Focus: (h + p, k)
Axis:
y=k
Vertex: (h, k)
p<0
p>0
共 y k兲2 4p共x h兲
(c) Horizontal axis: p > 0
(d) Horizontal axis: p < 0
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Appendix D.1
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Review of Algebra, Geometry, and Trigonometry
Equations of Ellipses 关Center: 冇h, k冈兴
y
y
(h, k)
(h, k)
2a
2b
2a
x
x
2b
共x h兲2 共 y k兲2
1
a2
b2
共x h兲2 共 y k兲2
1
b2
a2
Equations of Hyperbolas 关Center: 冇h, k冈兴
y
y
(h, k + c)
(h − c, k)
(h, k)
(h + c, k)
(h, k)
x
x
(h, k − c)
共x h兲2 共 y k兲2
1
a2
b2
共 y k兲2 共x h兲2
1
a2
b2
Solid Analytic Geometry
Distance Between 冇x1, y1, z1冈 and 冇x2 , y2 , z2冈
d 冪共x2 x1兲2 共 y2 y1兲2 共z2 z1兲2
Midpoint Between 冇x1, y1, z1冈 and 冇x2 , y2 , z2冈
Midpoint 冢
x1 x2 y1 y2 z1 z2
,
,
2
2
2
冣
Equation of Plane
Ax By Cz D 0
Equation of Sphere 关Center: 冇h, k, l冈, Radius: r兴
共x h兲2 共 y k兲2 共z l 兲2 r 2
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Properties and Measurement
Trigonometry
Definitions of the Six Trigonometric Functions
< 兾2
Right triangle definition: 0 <
opp
hyp
sin csc hyp
opp
adj
hyp
cos sec hyp
adj
opp
adj
tan cot adj
opp
Circular function definition:
y
r
x
cos r
y
tan x
sin
Hy
θ
Adjacent
is any angle in standard position and 共x, y) is a
point on the terminal ray of the angle.
r
y
r
sec x
x
cot y
y
csc
se
nu
te
po
Opposite
9781133105060_App_D1.qxp
r=
x2 + y2
(x, y)
r
y
θ
x
x
Signs of the Trigonometric Functions by Quadrant
Quadrant
sin
cos
tan
cot
sec
csc
I
II
III
IV
Trigonometric Identities
Reciprocal identities
1
csc
1
csc sin
sin
tan cos
sin
1
sec
1
sec cos
cos
cot sin
cos
tan
cot
1
cot
1
tan
Pythagorean identities
sin2
cos2
1
tan2
1 sec2
cot2
1 csc2
Reduction formulas
sin共 兲 sin
sin
sin共 兲
cos共 兲 cos
cos
cos共 兲
tan共 兲 tan
tan
tan共 兲
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Appendix D.1
Review of Algebra, Geometry, and Trigonometry
■
Trigonometry
(Continued)
Sum or difference of two angles
sin共 ± 兲 sin cos ± cos sin cos共 ± 兲 cos cos sin sin tan共 ± 兲 tan ± tan 1 tan tan sin共 兲 sin共 兲 sin2
sin2 cos共 兲 cos共 兲 cos2
sin2 Double-angle identities
sin 2 2 sin cos
cos 2 2 cos2
tan 2 1 1 2 sin2
2 tan
1 tan2
Multiple-angle identities
sin 3 3 sin
4 sin3
cos 3 3 cos
tan 3 4 cos3
3 tan tan3
1 3 tan2
sin 4 4 sin cos
cos 4 8 cos4
tan 4 8 sin3
8 cos2
cos
1
4 tan 4 tan3
1 6 tan2 tan4
Half-angle identities
sin2
cos2
1
共1 cos 2 兲
2
1
共1 cos 2 兲
2
Product identities
1
cos共
2
1
cos cos cos共
2
1
sin cos sin共
2
1
cos sin sin共
2
sin sin 1
cos共
2
1
兲 cos共
2
1
兲 sin共
2
1
兲 sin共
2
兲 兲
兲
兲
兲
D9
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Properties and Measurement
Library of Functions
Algebraic Functions
y
2
1
1
4
2
3
1
2
x
− 2 −1
−1
y
y
2
x
−2
1
−2
Linear or First-Degree
Polynomial
f 共x兲 x
1
y
2
3
1
3
2
1
2
x
−2
1
x
1
1
−1
2
x
−3
1 2 3
−2
2
Fourth-Degree Polynomial
f 共x兲 x 4
Fifth-Degree Polynomial
f 共x兲 x 5
Rational Function
1
x
f 共x兲 y
y
y
4
4
4
2
2
2
x
−2
Cubic or Third-Degree
Polynomial
f 共x兲 x 3
y
4
4
6
−2
2
−2
2
Quadratic or Second-Degree
Polynomial
f 共x兲 x 2
y
− 2 −1
−1
x
−2 −1
1
x
−4
4
−4
−2
x
2
−4
Rational Function
x1
x2
Rational Function
5
x2 4
f 共x兲 Rational Function
x2 1
x
f 共x兲 f 共x兲 y
y
4
2
3
1
x
2
−2 −1
1
x
1
2
3
4
Square Root Function
f 共x兲 冪x
1
2
−2
Cube Root Function
3 x
f 共x兲 冪
4
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Appendix D.1
D11
Review of Algebra, Geometry, and Trigonometry
■
Library of Functions (Continued)
Exponential and Logarithmic Functions
y
y
y
2
1
x
x
x
1
−1
2
3
4
−2
Exponential Function
f 共x兲 a x, a > 1
Exponential Function
f 共x兲 a x, 0 < a < 1
Logarithmic Function
f 共x兲 ln x
Trigonometric Functions
y
y
y
2
4
2
2
1
π
2
π
x
2π
π
2
−1
π
x
2π
−2
−2
Sine Function
f 共x兲 sin x
Cosine Function
f 共x兲 cos x
−π
Tangent Function
f 共x兲 tan x
y
y
4
x
π
y
4
4
2
2
−π
x
π
2
Cosecant Function
f 共x兲 csc x
−π
π
x
Secant Function
f 共x兲 sec x
−π
−π
2
π
2
π
x
Cotangent Function
f 共x兲 cot x
Nonelementary Functions
y
y
y
4
2
4
3
1
2
x
1
−2
x
−2 −1
1
ⱍⱍ
2
−2 −1
4
−2
2
Absolute Value Function
f 共x兲 x
x
Piecewise-Defined Function
冦
1 x,
x < 1
f 共x兲 冪x 1, x 1
Step Function
f 共x兲 冀x冁
1
2
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Properties and Measurement
D.2 Units of Measurements
■ Units of Measurement of Length
■ Units of Measurement of Area
■ Units of Measurement of Volume
■ Units of Measurement of Mass and Force
■ Units of Measurement of Temperature
■ Miscellaneous Units and Number Constants
Units of Measurement of Length
English System: Inch 冇in.冈, Foot 冇ft冈, Yard 冇yd冈, Mile 冇mi冈
1 mi ⫽ 5280 ft
1 yd ⫽ 3 ft
1 mi ⫽ 1760 yd
1 ft ⫽ 12 in.
Metric System: Millimeter 冇mm冈, Centimeter 冇cm冈, Meter 冇m冈, Kilometer 冇km冈
1 km ⫽ 1000 m
1 m ⫽ 1000 mm
1 m ⫽ 100 cm
1 cm ⫽ 10 mm
Conversion Factors 冇six significant figures冈
Metric to English
English to Metric
1 mm ⬇ 0.0393701 in.
1 cm ⬇ 0.393701 in.
1 m ⬇ 3.28084 ft
1 m ⬇ 1.09361 yd
1 km ⬇ 0.621371 mi
1 in. ⬇ 25.4000 mm
1 in. ⬇ 2.54000 cm
1 ft ⬇ 0.304800 m
1 yd ⬇ 0.914400 m
1 mi ⬇ 1.60934 km
Miscellaneous
1 fathom ⫽ 6 ft
1 astronomical unit ⬇ 93,000,000 mi (average distance between earth and sun)
1 light-year ⬇ 5,880,000,000,000 mi (distance traveled by light in 1 year)
Units of Measurement of Area
English System: Square Inch 冇in.2 冈, Square Foot 冇ft 2 冈, Square Yard 冇yd2 冈,
Acre, Square Mile 冇mi 2 冈
1 mi2 ⫽ 640 acres
1 yd2 ⫽ 9 ft2
1 acre ⫽ 43,560 ft2
1 ft2 ⫽ 144 in.2
Metric System: Square Centimeter 冇cm2 冈, Square Meter 冇m2 冈,
Square Kilometer 冇km2 冈
1 km2 ⫽ 1,000,000 m2
1 m2 ⫽ 10,000 cm2
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Appendix D.2
Units of Measurement of Area
■
Units of Measurements
(Continued)
Conversion Factors 冇six significant figures冈
Metric to English
English to Metric
1 cm2 ⬇ 0.155000 in.2
1 m2 ⬇ 10.7640 ft2
1 m2 ⬇ 1.19599 yd2
1 km2 ⬇ 0.386102 mi2
1 in.2 ⬇ 6.45160 cm2
1 ft2 ⬇ 0.0929030 m2
1 yd2 ⬇ 0.836127 m2
1 mi2 ⬇ 2.58999 km2
Miscellaneous
1 square mile ⫽ 1 section ⫽ 4 quarters
Units of Measurement of Volume
English System: Cubic Inch 冇in.3 冈 , Cubic Foot 冇ft 3 冈 , Cubic Yard 冇yd 3 冈 ,
Fluid Ounce 冇fl oz冈 , Pint 冇pt冈 , Quart 冇qt冈 , Gallon 冇gal冈
1 yd3 ⫽ 27 ft3
1 gal ⫽ 4 qt
1 qt ⫽ 32 fl oz
1 gal ⬇ 231 in.3
1 ft3 ⫽ 1728 in.3
1 qt ⫽ 2 pt
1 pt ⫽ 16 fl oz
1 ft3 ⬇ 7.48052 gal
Metric System: Cubic Centimeter 冇cm 3 or cc冈 , Cubic Meter 冇m3 冈 ,
Milliliter 冇mL冈 , Liter
1 m3 ⫽ 1000 liters
1 cm3 ⫽ 1 cc ⫽ 1 mL
1 liter ⫽ 1000 mL
Conversion Factors 冇six significant figures冈
Metric to English
1
⬇ 0.0610237
3
1 m ⬇ 35.3147 ft3
1 m3 ⬇ 1.30795 yd3
cm3
English to Metric
in.3
1 in.3 ⬇ 16.3871 cm3
1 ft3 ⬇ 0.0283168 m3
1 yd3 ⬇ 0.764555 m3
Miscellaneous
1 gallon ⫽ 5 fifths
1 fifth ⬇ 0.757 liter
1 quart ⫽ 4 cups
1 cup ⫽ 16 tablespoons
1 tablespoon ⫽ 3 teaspoons
1 barrel 共bl兲 ⫽ 42 gallons 共petroleum兲
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Properties and Measurement
Units of Measurement of Mass and Force
English System 冇Force or Weight冈 : Ounce 冇oz冈 , Pound 冇lb冈 , Ton
1 ton ⫽ 2000 lb
1 lb ⫽ 16 oz
Metric System 冇Mass冈 : Gram 冇g冈 , Kilogram 冇kg冈 , Metric Ton
1 metric ton ⫽ 1000 kg
1 kg ⫽ 1000 g
Conversion Factors 冇six significant figures at sea level冈
Metric to English
English to Metric
1 g ⬇ 0.0352740 oz
1 kg ⬇ 2.20462 lb
1 metric ton ⬇ 1.10231 ton
1 oz ⬇ 28.3495 g
1 lb ⬇ 0.453592 kg
1 ton ⬇ 0.907185 metric ton
Units of Measurement of Temperature
Fahrenheit 冇⬚F冈, Celsius 冇⬚C冈 , Kelvin or Absolute 冇K冈
Celsius to Fahrenheit
9
F ⫽ C ⫹ 32
5
Fahrenheit to Celsius
5
C ⫽ 共F ⫺ 32兲
9
Celsius to Kelvin
K ⫽ C ⫹ 273.15
Kelvin to Celsius
C ⫽ K ⫺ 273.15
Freezing temperature for water ⫽ 32⬚F ⫽ 0⬚C
Boiling temperature for water ⫽ 212⬚F ⫽ 100⬚C
Absolute zero temperature ⫽ 0 K ⫽ ⫺273.15⬚C ⫽ ⫺459.67⬚F
共0 K is, by definition, the lowest possible temperature, at which there
is no molecular activity.兲
Miscellaneous Units and Number Constants
␲ ⬇ 3.1415926535
e ⬇ 2.7182818284
Equatorial radius of the earth ⬇ 3963.19 mi ⬇ 6378.137 km
Polar radius of the earth ⬇ 3949.90 mi ⬇ 6356.752 km
Acceleration due to gravity at sea level ⬇ 32.1740 ft兾sec2
Speed of sound at sea level (standard atmosphere) ⬇ 1116.45 ft兾sec
Speed of light in vacuum ⬇ 186,282 mi兾sec
Density of water: 1 ft3 ⬇ 62.425 lb