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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