APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A1

D.1 Review of Algebra, Geometry, and TrigonometryAlgebra • Properties of Logarithms • Geometry • Plane Analytic Geometry • Solid Analytic Geometry •Trigonometry • Library of Functions

D Properties and Measurement

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.

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

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

3.

4.

5.

Quadratic Formula

Factors and Special Products

1.

2.

3.

4.

Factoring by Grouping

Binomial Theorem

1.

2.

3.

4.

5.

6.

7.

. . . � 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

ab � acad

�a�b � c�

ad�

b � cd

abac

�bc

a�bc

�a�bc�1

� �ab��

1c� �

abc

a�bc�d

� �ab��

dc� �

adbc

�ab��

cd� �

acbd

A2 APPENDIX D Properties and Measurement

Algebra (Continued)

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:

Series

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 �

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A3

�� < 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.

Geometry

Triangles

1. General triangle

Sum of triangles

Area (base)(height)

2. Similar triangles

3. Right triangle

(Pythagorean Theorem)

Sum of acute angles � � � 90�

c2 � a2 � b2

ab

�AB

�12

bh�12

� � � � 180�

logax �ln xln a

ln xy � y ln x

ln xy

� ln x � ln yln xy � ln x � ln y

ln e � 1ln 1 � 0

eln x � xln ex � x

A4 APPENDIX D Properties and Measurement

*

*�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!� . . . ,

β

α θh

b

b

aβ

α

b

ac

β

α

B

Aβ

α

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

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A5

Geometry (Continued)

4. Equilateral triangle

Height

Area

5. Isosceles right triangle

Area

Quadrilaterals (Four-Sided Figures)

1. Rectangle 2. Square

Area Area

3. Parallelogram 4. Trapezoid

Area Area

h

a

bh

b

a

h

b

�12

h�a � b�� bh

s

s

w

� �side�2 � s2� �length��width� � lw

�s2

2

��3s2

4

� h ��3s

2

s

s

45°

45°

60°

60° 60°

hs s

s

Circles and Ellipses

1. Circle 2. Sector of circle in radians

Area Area

Circumference

3. Circular ring 4. Ellipse

Area Area

Circumference

Solid Figures

1. Cone area of base

Volume

2. Right circular cone

Volume

Lateral surface area

3. Frustum of right circular cone

Volume

Lateral surface area � �s�R � r�

�� �r2 � rR � R2�h

3h

r

R

s

� �r�r2 � h2

��r2h

3

h

r

�Ah3

h

A

��A �

b

a

� 2��a2 � b2

2r

R

� �ab� � �R2 � r2�

r

s

θr

s � r � 2�r

� r2

2� �r2

��

A6 APPENDIX D Properties and Measurement

Geometry (Continued)

4. Right circular cylinder

Volume

Lateral surface area

5. Sphere

Volume

Surface area

Plane Analytic Geometry

Distance Between and

Midpoint Between and

Midpoint

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: y � bx � a

Ax � By � C � 0y � y1 � m�x � x1�

m1 � �1

m2

m1 � m2

m �y2 � y1

x2 � x1

�x2, y2��x1, y1�

� �x1 � x2

2,

y1 � y2

2 ��x2, y2��x1, y1�

d � ��x2 � x1�2 � �y2 � y1�2

�x2, y2��x1, y1�

� 4�r2

�43

�r3 r

� 2�rh

� �r2hh

r

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A7

Equations of Parabolas Vertex: h, k

(a) Vertical axis: (b) Vertical axis: (c) Horizontal axis: (d) Horizontal axis:

Equations of Ellipses Center:

�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�h, k�

p < 0p > 0p < 0p > 0

�y � k�2 � 4p�x � h��x � h�2 � 4p�y � k�

p < 0

AxisFocus

Vertex

Directrix

p > 0

Axis:y = k

Focus: (h + p, k)

Vertex: (h, k)

x = h − pDirectrix:

p < 0

Focus

VertexDirectrix

Axis

p > 0

x = h

Focus:(h, k + p)

Vertex:( , )h k

Directrix:y = k − p

Axis:

��

A8 APPENDIX D Properties and Measurement

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

Standard form:

General form: Ax2 � Ay2 � Dx � Ey � F � 0

�x � h�2 � �y � k�2 � r2

��

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A9

Solid Analytic Geometry

Distance Between and

Midpoint Between and

Midpoint

Equation of Plane

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

Trigonometry

Definitions of the Six Trigonometric Functions

Right triangle definition:

cot �adj.opp.

tan �opp.adj.

sec �hyp.adj.

cos �adj.hyp.

csc �hyp.opp.

sin �opp.hyp.

0 < < ��2

�x � h�2 � �y � k�2 � �z � l�2 � r2

��

Ax � By � Cz � D � 0

� �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�

Adjacent

Opp

osite

Hypotenuse

θ

Equations of Hyperbolas Center: h, k

�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��

Plane Analytic Geometry (Continued)

A10 APPENDIX D Properties and Measurement

Circular function definition: is any angle and is a point 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)

x

(x, y)

x

yr

θ

r = x2 + y2

y

Quadrant sin cos tan cot sec csc

I � � � � � �

II � � � � � �

III � � � � � �

IV � � � � � �

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 �

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A11

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

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� � x5f�x� � x4

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

A12 APPENDIX D Properties and Measurement

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 Compound 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 π2

ππ2

−−x

ππ−

4

y

x

4

2

y

ππ2

−

f�x� � tan xf �x� � cos xf�x� � sin x

xππ−

4

2

y

x

−2

−1

2

y

π π π2

2x

π π π2

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

APPENDIX D.1 Review of Algebra, Geometry, and Trigonometry A13