D.1 Review of Algebra, Geometry, and Trigonometry€¦ · Plane Analytic Geometry Distance Between...

Appendix D.1 ■ Review of Algebra, Geometry, and Trigonometry D1

■ Algebra

■ Properties of Logarithms

■ Geometry

■ Plane Analytic Geometry

■ Solid Analytic Geometry

■ Trigonometry

■ Library of Functions

Algebra

Operations with Exponents

1. 2. 3.

4. 5. 6.

7. 8.

Exponents and Radicals (n and m are positive integers)

1. 2.n factors

3. *4.

5. 6.

7.

Operations with Fractions

1.

2.

3.

4. 5.

* If n is even, the principal nth root is defined to be positive.

ab � acad

�a�b � c�

ad�

b � cd

a�bc

�a�bc�1

� �ab��

1c� �

abc

abac

�bc

a�bc�d

� �ab��

dc� �

adbc

�ab��

cd� �

acbd

ab

�cd

�ab �

dd� �

cd �

bb� �

adbd

�bcbd

�ad � bc

bd

ab

�cd

�ab �

dd� �

cd �

bb� �

adbd

�bcbd

�ad � bc

bd

2�x � �x

xm�n � �xm�1�n � n�xm

xm�n � �x1�n�m � � n�x�mx1�n � n�x

x � ann�x � ax � 0x�n �1xn ,

x � 0x0 � 1,xn � x � x � x . . . x

xnm� x�nm�cxn � c�xn�

�xn � ��xn��xn�m � xnm�xy�

n

�xn

yn

�xy�n � xnynxn

xm � xn�mxnxm � xn�m

D Properties and Measurement

D.1 Review of Algebra, Geometry, andTrigonometry

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Quadratic Formula

Factors and Special Products

1.

2.

3.

4.

Factoring by Grouping

Binomial Theorem

1.

2.

3.

4.

5.

6.

7.

8.

Miscellaneous

1. If then or

2. If and then

3. Factorial: etc.

Sequences

1. Arithmetic:

2. Geometric:

3. General harmonic:

4. Harmonic:

5. p-Sequence:11p ,

12p ,

13p ,

14p ,

15p , . . .

11

, 12

, 13

, 14

, 15

, . . .

1a

, 1

a � b,

1a � 2b

, 1

a � 3b,

1a � 4b

, 1

a � 5b, . . .

ar0 � ar1 � ar2 � ar3 � . . . � arn �a�1 � rn�1�

1 � r

ar0, ar1, ar2, ar3, ar4, ar5, . . .

a, a � b, a � 2b, a � 3b, a � 4b, a � 5b, . . .

0! � 1, 1! � 1, 2! � 2 � 1, 3! � 3 � 2 � 1, 4! � 4 � 3 � 2 � 1,

a � b.c � 0,ac � bc

b � 0.a � 0ab � 0,

. . . ± nan�1x � an�x � a�n � xn � naxn�1 �n�n � 1�

2!a2xn�2 �

n�n � 1��n � 2�3!

a3xn�3 �

. . . � nan�1x � an�x � a�n � xn � naxn�1 �n�n � 1�

2!a2xn�2 �

n�n � 1��n � 2�3!

a3xn�3 �

�x � a�4 � x4 � 4ax3 � 6a2x2 � 4a3x � a4

�x � a�4 � x4 � 4ax3 � 6a2x2 � 4a3x � a4

�x � a�3 � x3 � 3ax2 � 3a2x � a3

�x � a�3 � x3 � 3ax2 � 3a2x � a3

�x � a�2 � x2 � 2ax � a2

�x � a�2 � x2 � 2ax � a2

acx3 � adx2 � bcx � bd � ax2�cx � d� � b�cx � d� � �ax2 � b��cx � d�

x4 � a4 � �x � a��x � a��x2 � a2�

x3 � a3 � �x � a��x2 � ax � a2�

x3 � a3 � �x � a��x2 � ax � a2�

x2 � a2 � �x � a��x � a�

x ��b ± �b2 � 4ac

2aax2 � bx � c � 0

D2 Appendix D ■ Properties and Measurement

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Series

*

*�1 < x < 1 �1 � x��k � 1 � kx �k�k � 1�x2

2!�

k�k � 1��k � 2�x3

3!�

k�k � 1��k � 2��k � 3�x4

4!� . . . ,

�1 < x < 1 �1 � x�k � 1 � kx �k�k � 1�x2

2!�

k�k � 1��k � 2�x3

3!�

k�k � 1��k � 2��k � 3�x4

4!� . . . ,

�� < x < � cos x � 1 �x2

2!�

x4

4!�

x6

6!� . . . ,

�� < x < � sin x � x �x3

3!�

x5

5!�

x7

7!� . . . ,

�� < x < � ex � 1 � x �x2

2!�

x3

3!�

x4

4!�

x5

5!� . . . �

xn

n!� . . . ,

0 < x 2 ln x � �x � 1� ��x � 1�2

2�

�x � 1�3

3�

�x � 1�4

4� . . . �

��1�n�1�x � 1�n

n� . . . ,

�1 < x < 1 1

1 � x� 1 � x � x2 � x3 � x4 � x5 � . . . � ��1�nxn � . . . ,

0 < x < 2 1x

� 1 � �x � 1� � �x � 1�2 � �x � 1�3 � �x � 1�4 � . . . � ��1�n�x � 1�n � . . . ,

Properties of Logarithms

Inverse Properties

1.

2.

Properties of Logarithms

1.

2.

3.

4.

5.

6.

*The convergence at depends on the value of k.x � ±1

logax �ln xln a

ln xy � y ln x

ln xy

� ln x � ln y

ln xy � ln x � ln y

ln e � 1

ln 1 � 0

eln x � x

ln ex � x


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Geometry

Triangles

1. General triangle

Sum of angles

Area (base)(height)

2. Similar triangles

3. Right triangle

(Pythagorean Theorem)Sum of acute angles

4. Equilateral triangle

5. Isosceles right triangle

Quadrilaterals (Four-Sided Figures)

1. Rectangle 2. Square

3. Parallelogram 4. Trapezoid

h

a

bh

b

a

h

b

Area �12

h�a � b�Area � bh

s

s

w

Area � �side�2 � s2Area � �length��width� � lw

Area �s2

2

Area ��3s2

4

Height � h ��3s

2

� � � 90�c2 � a2 � b2

ab

�AB

�12

bh�12

� � � � 180�


β

α θ h

b

b

aβ

α

b

ac

β

α

B

Aβ

α

s

s

45°

45°

60°

60° 60°

hs

s

s

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Circles and Ellipses

1. Circle 2. Sector of circle in radians

3. Circular ring 4. Ellipse

Solid Figures

1. Cone area of base

2. Right circular cone

3. Frustum of right circular cone

4. Right circular cylinder

5. Sphere

Surface area � 4�r2

Volume �43

�r3

Lateral surface area � 2�rh

Volume � �r2h

Lateral surface area � �s�R � r�

Volume �� r2 � rR � R2�h

3

Lateral surface area � �r�r2 � h2

Volume ��r2h

3

Volume �Ah3

��A �

b

a

Circumference � 2��a2 � b2

2r

R

Area � �abArea � � �R2 � r2�

r

s

θ r

s � r Circumference � 2�r

Area � r2

2Area � �r2

��


h

A

h

r

h

r

R

s

h

r

r

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Plane Analytic Geometry

Distance Between and

Midpoint Between and

Slope of Line Passing Through and

Slopes of Parallel Lines

Slopes of Perpendicular Lines

Equations of Lines

Point-slope form: General form:Vertical line: Horizontal line:Slope-intercept form:

Equations of Circles Center: h, k , Radius: r

Standard form:General form:

Equations of Parabolas Vertex: h, k

(a) Vertical axis: (b) Vertical axis:

(c) Horizontal axis: (d) Horizontal axis: p < 0p > 0

�y � k�2 � 4p�x � h�

p < 0

AxisFocus

Vertex

Directrix

p > 0

Axis:y = k

Focus: (h + p, k)

Vertex: (h, k)

x = h − pDirectrix:

p < 0p > 0

�x � h�2 � 4p�y � k�

p < 0

Focus

VertexDirectrix

Axis

p > 0

x = h

Focus:(h, k + p)

Vertex:( , )h k

Directrix:y = k − p

Axis:

��

Ax2 � Ay2 � Dx � Ey � F � 0�x � h�2 � �y � k�2 � r2

��

y � mx � by � bx � a

Ax � By � C � 0y � y1 � m�x � x1�

m1 � �1

m2

m1 � m2

m �y2 � y1

x2 � x1

x2, y2�x1, y1�

Midpoint � �x1 � x2

2,

y1 � y2

2 �x2, y2�x1, y1�

d � ��x2 � x1�2 � �y2 � y1�2

x2, y2�x1, y1�


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Equations of Ellipses Center: h, k

Equations of Hyperbolas Center: h, k

Solid Analytic Geometry

Distance Between and

Midpoint Between and

Equation of Plane

Equation of Sphere Center: h, k, l , Radius: r

�x � h�2 � �y � k�2 � �z � l�2 � r2

��

Ax � By � Cz � D � 0

Midpoint � �x1 � x2

2,

y1 � y2

2,

z1 � z2

2 �x2, y2, z2�x1, y1, z1�

d � ��x2 � x1�2 � �y2 � y1�2 � �z2 � z1�2

x2, y2, z2�x1, y1, z1�

�y � k�2

a2 ��x � h�2

b2 � 1�x � h�2

a2 ��y � k�2

b2 � 1

x

(h, k + c)

(h, k − c)

(h, k)

y

x

(h − c, k) (h + c, k) (h, k)

y

��

�x � h�2

b2 ��y � k�2

a2 � 1�x � h�2

a2 ��y � k�2

b2 � 1

x

(h, k) 2a

2b

y

x

(h, k) 2b

2a

y

��


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Trigonometry

Definitions of the Six Trigonometric Functions

Right triangle definition:

Circular function definition: is any angle in standard position and is apoint on the terminal ray of the angle.

Signs of the Trigonometric Functions by Quadrant

Trigonometric Identities

Reciprocal identities

Pythagorean identities

Reduction formulas

tan � tan� � ��cos � �cos� � ��sin � �sin� � ��

tan�� tan cos�� cos sin�� sin

cot2 � 1 � csc2 tan2 � 1 � sec2 sin2 � cos2 � 1

cot � cos sin

tan �sin cos

cot �1

tan sec �

1cos

csc �1

sin

tan �1

cot cos �

1sec

sin �1

csc

cot �xy

tan �yx

sec �rx

cos �xr

csc �ry

sin �yr

�x, y)

cot �adjopp

tan �oppadj

sec �hypadj

cos �adjhyp

csc �hypopp

sin �opphyp

0 < < ��2


Adjacent

Opp

osite

Hypotenuse

θ

x

(x, y)

x

yr

θ

r = x2 + y2

y

Quadrant sin cos tan cot sec csc

I � � � � � �

II � � � � � �

III � � � � � �

IV � � � � � �

9781133109280_App_D1.qxp 12/3/11 1:34 PM Page D8

Trigonometry (Continued)

Sum or difference of two angles

Double-angle identities

Multiple-angle identities

Half-angle identities

Product identities

cos sin � �12

sin� � �� 12

sin� � ��

sin cos � �12

sin� � �� 12

sin� � ��

cos cos � �12

cos� � �� 12

cos� � ��

sin sin � �12

cos� � �� 12

cos� � ��

cos2 �12

�1 � cos 2 �

sin2 �12

�1 � cos 2 �

tan 4 �4 tan � 4 tan3

1 � 6 tan2 � tan4

cos 4 � 8 cos4 � 8 cos2 � 1

sin 4 � 4 sin cos � 8 sin3 cos

tan 3 �3 tan � tan3

1 � 3 tan2

cos 3 � �3 cos � 4 cos3

sin 3 � 3 sin � 4 sin3

tan 2 �2 tan

1 � tan2

cos 2 � 2 cos2 � 1 � 1 � 2 sin2

sin 2 � 2 sin cos

cos� � �� cos� � �� cos2 � sin2 �

sin� � �� sin� � �� sin2 � sin2 �

tan� ± �� tan ± tan �

1 � tan tan �

cos� ± �� cos cos � � sin sin �

sin� ± �� sin cos � ± cos sin �


9781133109280_App_D1.qxp 12/3/11 1:34 PM Page D9

Library of Functions

Algebraic Functions

Linear or First-Degree Quadratic or Second-Degree Cubic or Third-DegreePolynomial Polynomial Polynomial

Fourth-Degree Polynomial Fifth-Degree Polynomial Rational Function

Rational Function Rational Function Rational Function

Square Root Function Cube Root Functionf�x� � 3�xf�x� � �x

x−2 −1 1 2

−2

1

2

y

x1 2 3 4

1

2

3

4

y

f�x� �x2 � 1

xf�x� �

5x2 � 4

f�x� �x � 1x � 2

x42−4 −2

2

4

y

x4−4

2

4

y

x−2 4 6

2

4

−2

−4

y

f�x� �1x

f�x� � x5f�x� � x4

x

y

−3 1 2 3

1

2

3

x−2 1 2

−2

−1

1

2

y

x−2 −1 1 2

1

2

3

4

y

f�x� � x3f�x� � x2f�x� � x

x−2 1 2

−2

−1

1

2

y

x−2 −1 1 2

1

2

3

4

y

x−2 −1 1 2

−2

−1

1

2

y


9781133109280_App_D1.qxp 12/3/11 1:34 PM Page D10

Library of Functions (Continued)

Exponential and Logarithmic Functions

Exponential Function Exponential Function Logarithmic Function

Trigonometric Functions

Sine Function Cosine Function Tangent Function

Cosecant Function Secant Function Cotangent Function

Nonelementary Functions

Absolute Value Function Piecewise-Defined Function Step Function

f�x� � x�f�x� � �1 � x,�x � 1,

x < 1 x � 1

f�x� � �x�

x−2 −1 21

1

2

y

x42−2

−2

4

y

x−2 −1 1 2

1

2

3

4

y

f�x� � cot xf�x� � sec xf�x� � csc x

x

4

2

y

π π 2

π π 2

− − x

π π −

4

y

x

4

2

y

π π 2

−

f�x� � tan xf�x� � cos xf�x� � sin x

xπ π −

4

2

y

x

−2

−1

2

y

π π π 2

2x

π π π 2

2

2

−2

1

y

f�x� � ln xf�x� � ax, 0 < a < 1f�x� � ax, a > 1

x1 2 3 4

1

2

−2

−1

y

x

y

x

y


9781133109280_App_D1.qxp 12/3/11 1:34 PM Page D11


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 in. , Foot ft , Yard yd , Mile mi

Metric System: Millimeter mm , Centimeter cm , Meter m , Kilometer km

Conversion Factors six significant figures

Metric to English English to Metric

Miscellaneous

1 fathom

1 astronomical unit (average distance between earth and sun)

1 light-year (distance traveled by light in 1 year)

Units of Measurement of Area

English System: Square Inch in. , Square Foot ft , Square Yard yd ,Acre, Square Mile mi

Metric System: Square Centimeter cm , Square Meter m , Square Kilometer km

1 m2 � 10,000 cm21 km2 � 1,000,000 m2

�2��2��2�

1 ft2 � 144 in.21 yd2 � 9 ft21 acre � 43,560 ft21 mi2 � 640 acres

�2��2��2��2�

� 5,880,000,000,000 mi

� 93,000,000 mi

� 6 ft

1 mi � 1.60934 km1 km � 0.621371 mi

1 yd � 0.914400 m1 m � 1.09361 yd

1 ft � 0.304800 m1 m � 3.28084 ft

1 in. � 2.54000 cm1 cm � 0.393701 in.

1 in. � 25.4000 mm1 mm � 0.0393701 in.

��

1 cm � 10 mm1 m � 1000 mm

1 m � 100 cm1 km � 1000 m

��

1 ft � 12 in.1 yd � 3 ft

1 mi � 1760 yd1 mi � 5280 ft

��

9781133109280_App_D2.qxp 12/3/11 1:36 PM Page D12

Units of Measurement of Area (Continued)



Miscellaneous

1 square mile section quarters

Units of Measurement of Volume

English System: Cubic Inch in. , Cubic Foot ft , Cubic Yard yd , Fluid Ounce fl oz , Pint pt , Quart qt , Gallon gal

Metric System: Cubic Centimeter cm or cc , Cubic Meter m , Milliliter mL , Liter



Miscellaneous

1 barrel �bl� � 42 gallons �petroleum�1 tablespoon � 3 teaspoons

1 cup � 16 tablespoons1 quart � 4 cups

1 fifth � 0.757 liter1 gallon � 5 fifths

1 yd3 � 0.764555 m31 m3 � 1.30795 yd3

1 ft3 � 0.0283168 m31 m3 � 35.3147 ft31 in.3 � 16.3871 cm31 cm3 � 0.0610237 in.3

��

1 cm3 � 1 cc � 1 mL

1 liter � 1000 mL1 m3 � 1000 liters

��3��3�

1 ft3 � 7.48052 gal1 gal � 231 in.31 pt � 16 fl oz1 qt � 32 fl oz

1 qt � 2 pt1 gal � 4 qt

1 ft3 � 1728 in.31 yd3 � 27 ft3

��3��3��3�

� 4� 1

1 mi2 � 2.58999 km21 km2 � 0.386102 mi21 yd2 � 0.836127 m21 m2 � 1.19599 yd2

1 ft2 � 0.0929030 m21 m2 � 10.7640 ft21 in.2 � 6.45160 cm21 cm2 � 0.155000 in.2

��

Appendix D.2 ■ Units of Measurements D13

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Units of Measurement of Mass and Force

English System Force or Weight : Ounce oz , Pound lb , Ton

Metric System Mass : Gram g , Kilogram kg , Metric Ton

Conversion Factors six significant figures at sea level


Units of Measurement of Temperature

Fahrenheit F , Celsius C , Kelvin or Absolute K

Celsius to Fahrenheit Fahrenheit to Celsius

Celsius to Kelvin Kelvin to Celsius

Freezing temperature for water

Boiling temperature for water

Absolute zero temperature

is, by definition, the lowest possible temperature, at which there is no molecular activity.

Miscellaneous Units and Number Constants

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

Speed of sound at sea level (standard atmosphere) 1116.45

Speed of light in vacuum 186,282

Density of water: 1 ft3 � 62.425 lb

mi�sec�ft�sec�

ft�sec2��

��

e � 2.7182818284� � 3.1415926535

��0 K

� 0 K � �273.15�C � �459.67�F

� 212�F � 100�C

� 32�F � 0�C

C � K � 273.15K � C � 273.15

C �59

�F � 32�F �95

C � 32

��

1 ton � 0.907185 metric ton1 metric ton � 1.10231 ton

1 lb � 0.453592 kg1 kg � 2.20462 lb

1 oz � 28.3495 g1 g � 0.0352740 oz

��

1 kg � 1000 g1 metric ton � 1000 kg

��

1 lb � 16 oz1 ton � 2000 lb

��


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