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A C A D E M I C S K I L L S C E N T R E ( A S C )A C A D E M I C S K I L L S C E N T R E ( A S C )
P e t e r b o r o u g h • O s h a w a
Formula for Success a Mathematics Resource
Phone (705) 748-1720 Fax (705) 748-1830 e-mail: [email protected]/academicskills
Section 1: Formulas and Quick Reference Guide
1. Formulas From Geometry 12. Interval Notation 33. Coordinate Geometry 44. Steps in completing the Square 55. The Graphs of the Eight Basic Functions 66. Radian Measure 87. Special Angles 98. The Unit Circle 109. Inverse Trig Functions 1110. Trigonometric Identities 1211. Sine Law 1312. Cosine Law 1413. Graphs of Trig Functions 15
Section 2: Mathematics Review Modules
1. Order of Operations and Substitution. 162. Fractions, Decimals and Percent 203. Exponents, Roots and Factorials 27
Section 3: Calculus/Algebra Prep
1. Algebrai) Terminology 32ii) Expanding and Factoring 33iii) Factor Theorem and Synthetic Division 38
2. Equationsi) Definitions 40ii) Linear Equations 41iii) X and Y intercepts 42iv) Solving Quadratics Using Factoring 43v) Solving Quadratics Using Quadratic Formula 44vi) Solving Absolute Value Equations 46vii) Solving Polynomial Equations 47
3. Exponentsi) Exponents 49ii) The Exponential Function and its Inverse the Logarithmic Function 51iii) The Common Logarithm 53iv) The Natural Logarithm 53v) Laws of Logarithms 54vi) Solving Equations with Exponents or Logarithms 56vii) Expoential Growth and Decay 58
4. Functions and Transformationsi) Definition of a function 60ii) Function Notation 61 5. Inequalitiesi) Inequalities Definition 62ii) Linear Inequalities 62iii) Polynomial Inequalities 63
6. Trigonometric Functionsi) Trig Definitons 66ii) CAST Rule 68iii) Radian Measure 70iv) Finding Exact Values of Trig Ratios 71v) Using a calculator to find Trig Ratios 74vi) Trig Equations 76vii) Trig Identities 77viii) Graphs of y = sinx and y = cosx 80
The Academic Skills Centre would like to thank the following for their contributions to this publication: Ruth Brandow, Ellen Dempsey, Marj Tellis, Lisa Davies.
For information on this or any of our services, contact the Academic Skills Centre, Trent University, Peterborough, Ontario K9J 7B8
Contents
Section 1Formulas and Quick Reference Guide 1. Formulas from GeometryPythagoream Theorem
Triangle with Base and Height Given
Triangle with Base and Height Unknown
Area of a Parallelogram
Area of a Trapezoid
1
If a triangle has sides a, b, and c (c is the hypotenuse), then c2 = a2 + b2
The area of a triangle whose base is b and altitude is h (the perpendicular distance to the base) is A = ½ b x h
The area of a triangle with angles A, B, and C and sides opposite a, b, and c, respectively:
Area = ½ ab x sinC
The area of a parallelogram with base b and altitude h is Area = bh
The area of a trapezoid whose parallel sides are a and b and altitude is h is Area = ½(a + b)h
2
Circle Formulae
3 Dimensional Solids
Sphere
Right Circular Cylinder
Right Circular Cone
For a circle whose radius is r and diameter is d (d = 2r): Circumference : C = 2pr= pd Area: A = pr2
If s is the length of an arc subtended by a central angle of q radians, then s = rq
A Sphere with radius r has: Surface Area = 4pr2 Volume = 4/3 pr3
A right circular cylinder where r is the radius of the base and h is the altitude has:
Surface area = 2prh Volume = pr2h
A right circular cone where r is the radius of the base and h is the altitude has:
Surface area = p r2+ h2
Volume = 1/3 pr2h
3
2. Interval Notation
}{ ≤≤
}{ <≤
}{ ≤<
∞
∞
∞ ≥
∞ ≥
∞ ∞
4
3. Co-ordinate Geometry
=
=
−−
−+−
+
++
−
5
4. Steps in Completing the Square
++=++
+
−
++=++
−
+
+=
−+
+=
++
++=+−++=++
−−=
−−=
+−−−+−=+−=+−
−
−−=
+−−=
+−−+−−=+−−
6
5. The Graphs of the Eight Basic Functions
7
5. The Graphs of the Eight Basic Functions
8
ππππππ
ππππ
π
π
ππ=
ππ
=
6. Trigonometry - Radian Measure
9
6. Trigonometry - Radian Measure 7. Special AnglesFinding The Exact Value of a Trigonometric Ratio
10
8. The Unit Circle
11
9. Inverse Trigonometric Functions(a.k.a. arcsin, arccos and arctan)
−
ππ
≤≤− ππ
≤≤−
[ ]π ≤≤− π≤≤
π
π
π
−
−
π
12
10. Trigonometric Identities
=
−+
+−
−
13
11. Sine Law
14
12. Cosine Law
−+
15
13. Graphs of Trigonometric Functions
16
Section 2Mathematics Review Modules
1. Order of Operations and SubstitutionOrder of Operations
+ − − ÷ +
+ − ÷ += + − ÷ += + − += + += +=
16214
241824810
23128102312)4(210
=+=
+−=+−+=
+÷−+=+÷−+
17
+
+
= =
/ +/
=
18
Substituting into Formulae
+
+
=+
=
=
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
19
20
2. Fractions, Decimals, and Percents
Includes Rounding and Significant Digits
Fractions
• • •
÷ =
=
• • •
×
=
×
=
=
= =
21
2. Fractions, Decimals, and Percents
×
=
×
=
=
= =
÷
= ×
= =
+ =+
÷ = =
+ =+
+ = + =+
+ = + = + =
22
÷
= ×
= =
+ =+
÷ = =
+ =+
+ = + =+
+ = + = + =
23
Percents
24
Decimals
÷ =
÷
÷
25
Rounding and Significant Digits
• •
26
• • •
• •
•
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
27
3. Powers, Roots and Factorials
Powers
28
29
= + = =
=
+ ≠ +
=
=
Roots
30
⇒ ⋅ =
÷ = − ⇒ ÷ = −
( ) = ⇒ = ⋅ =
= ( )⇒ = =
= = − ⇒
=
= ( )⇒ − =
⇒− =−
− =⇒ = =−
= ⇒ = =
= ⇒ = = =
⇒ = = =−
−
Laws of Exponents
31
=
/ / / / // / / / /
==
=⋅
=
=
FactorialsLaws of Exponents
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
32
Section 3 Calculus/Algebra Prep
1. Essential Algebra
i) Terminology
− +
−
− +
−
− +
33
−
+ − −
+ −
− +
( ) − +
− +
−
− +
−
− +
− +
−
− +
−
− +
+ + = ≠
ii) Expanding and Factoring
34
−
+ − −
+ −
− +
( ) − +
35
−
− = − + = + − +
+ + + + − −∴ − − = − +
+ −
∴ + − = + −
36
+ + + + ≠ ≠
+ −
∴
+ − = + − −
+ + + + ≠ ≠
+ −
∴
+ − = + − −
+ + ( ) +
− + ( ) −
( )
37
+ −
+ = + − + − = − + +
− = − + +
+
= +
− +
38
iii) Factor Theorem(polynomial of degree three or greater)
+ − + ÷ −
− = ∴ =
∴
+ + + − + = − + + +
− + −
∴ − − − + − − =
39
− − − +
∴ − − − ÷ + − −
− + − + + +
− + + − + +
∴ + − +
− − − +
∴ − − − ÷ + − −
− + − + + +
− + + − + +
∴ + − +
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
40
2. Equations of Straight Lines
i) Definitions
−
−
− + −
41
∴
−−
+=
( )
−
−
+
=
∴
−−
+=
( )
−
−
+
=
Equations
ii) Linear Equations
42
iii) X and Y intercepts
∴
≠
∴
43
iv) Quadratic Equations Using Factoring
− + = ∴ ∴
= − + = ∴∴
+ − + = + − + − − + = + − + − − = ∴ ∴ −
44
v) Solving Quadratics Using Quadratic Formula
=− ± −
( )
( )
( )
+ + =
+ + − = −
+ − = −
+ =
+ =
+ = ±
−
−
−
− ± −=
− − =
= ± − − ∴ = ±
∴ = ± ∴ = ±
∴ = ±
45
− = −
= ± − ∴ = ± −
− ∴ = ±
− − =
− = − − =
46
vi) Absolute Value Equations
=
= −
= ⇒ ≥ = < = −
+ =
⇒ ∴ ⇒ ∴
− = −
⇒ ∴ = −
≥
⇒ ⇒ ∴ =
<
47
vii) Solving Polynomial Equations(With Degree Three or More)
+ + =
+ + =+ + =
+ =
+ − + =
= − = ± −
=
=
± −
±
48
− + =
= − + ∴
= − + −
∴ = − + − ∴ − + − =
∴ = = − =
− + = − − = ∴ − + − + =
∴ = ± = ±
∴ ∴ ∴ ∴
∴ − + + = ∴ − − + − + − − + − =
∴ − − + − = = −
[ ]∴ − − + − − =
∴ − + − =
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
49
3. Exponentsi) Laws of Exponents
≠
= + ⇒ ⋅ =
÷ = − ⇒ ÷ = −
= ⇒ = ⋅ =
= ⇒ = =×
( )
= = − ( )⇒ =
= ⇒ − =
⇒ − = −
− = ⇒ = =−
= ⇒ = =
= ⇒ = = =
⇒ = = =− −
−
−=
50
− −
= −
− − = − − = −
− =
−
−
=−
−
=
−
=−
=
−
=
( )=
=
( ) −
−
= − −
− −
= −
− − = − − = −
− =
−
−
=−
−
=
−
=−
=
−
=
( )=
=
( ) −
−
= − −
=
( )
( )
−−
= −−
−
( )
=−
=
=
=
= =
=
= ⇒ =
6.
51
=
( )
( )
−−
= −−
−
( )
=−
=
=
=
= =
=
= ⇒ =
=
( )
( )
−−
= −−
−
( )
=−
=
=
=
= =
=
= ⇒ =
ii) The Exponential Function and Its Inverse The Logarithmic Function
52
= = = =
=
=
= = = =
= =
=
= =
= −
( ) −=
= =
= =
− =
= −
= =
53
iii) The Common Logarithm
iv) The Natural Logarithm (ln and e)
=
=
− = =
=
v) Laws of Logarithms
54
= + ⇒ = +
= − ( )⇒ = −
= ⇒ =
= ⇒ =
= ⇒ =
+
[ ]= = −
( )=
= =
=
=
=
55
=
= = +
=
=
=
( )
= − =
=
≈
=
≈
56
vi) Solving Equations with Exponents or Logarithms
=
=
≈
+ − − =
( )
+− =
= +
−
= +
=
= ±
= ≠ −
+ + = + =
= + + − =
∴
57
= =
− = − = − =
− =
≈
− =
− − =
= = −
− − = − − =
− + =
= = − ∴
58
vii) Exponential Growth and Decay
÷
=
≈
∴
59
= +
≈∴
=
≈
∴
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
4. Functions and Transformationsi) Definition of a Function
60
ii) Function Notation
= ≠−
= −
− = − −−
= ≠− −−
∴
61
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
62
5. Inequalitiesi) Inequalities Definition
≥ ≤
ii) Linear Inequalities
≤
− ∞
63
ii) Polynomial Inequalities(Quadratic, Cubic and Quartic Inequalities)
− > − − >
− ∞ − ∞ ∴
− ∞ − ∞
− ≤ + + − − ≤
−
− ∞ −
− [ ]−
∞
∴ − [ ]−
−
64
⇒
− ∞ − ∞
∴ ∞
− < − <
− =
⇒ ( )− ∞ ( )∞
∴ ≠ ( )− ∞
− + > ( )( ) − − >
⇒ = ± = ± ( )− ∞ − ( )∞
∴ ( ) ( ) ( )− ∞ − − ∞
⇒
− ∞ − ∞
∴ ∞
− < − <
− =
⇒ ( )− ∞ ( )∞
∴ ≠ ( )− ∞
− + > ( )( ) − − >
⇒ = ± = ± ( )− ∞ − ( )∞
∴ ( ) ( ) ( )− ∞ − − ∞
⇒
− ∞ − ∞
∴ ∞
− < − <
− =
⇒ ( )− ∞ ( )∞
∴ ≠ ( )− ∞
− + > ( )( ) − − >
⇒ = ± = ± ( )− ∞ − ( )∞
∴ ( ) ( ) ( )− ∞ − − ∞
Polynomial Inequalities(With Non Real Roots)
65
( )( ) − + + ≤
( ) + + ≤ −
⇒ = ± ( ] [ ] [ )− ∞ − − ∞
∴
-4 4
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
66
6. Trigonometric Functionsi) Definitions
° °
° + =
= =
= =
= =
°
− °
67
6. Trigonometric Functionsi) Definitions
= =
=
=
= =
= =
° ° °
°
68
ii) CAST Rule
°
° ° − ° °
+ =
+ − =
69
−
−
∴
+ =
+ = ±
∴ = ±
70
iii) Radian Measure
θ =
θ °
π π θ π=
°
∴ ° = π
∴ ° =°
π
=
°π
°
° − °
( )π
π
π( )π
−
π
π
−π
( )
ππ = °
π ( )− = − °π
π
( ) π π= = °
71
iv) Finding Exact Values Of Trig RatiosSpecial Angles And The Unit Circle
( )π
° ( )π ° π
°
=
=
72
θ π= ° ( )
= = × =
π
π
π
π ∴ = −
π
73
π6
π4
π3
π π= −
−
π
=
π/2 (0,1) ∴ = π
π (−1,0)0 (1,0) ∴ = π
π
2π
π − = − π
π π
π
π
= = −
π
π
− = =π π
π
π π= −
π ππ− =
= −
π
74
v) Using The Calculator To Find The Ratios
−
°
° ° −
2π
π
π
π π
π
−
π
= π
π
π
π
π
− − −
°
( ) − = ° − °
°
≈ ° °
−
− − = °
° °
2π
− − π − 1.27 π + 1.27 π≈ π.≈
75
−
π
= π
π
π
π
π
− − −
°
( ) − = ° − °
°
≈ ° °
−
− − = °
° °
2π
− − π − 1.27 π + 1.27 π≈ π.≈
76
vi) Trigonometric Equations
≤ ≤ π
2π −
π
π
π π
= π
2π
− − = − − =
∴ = −
π
π π
π
+ − =
∴ −
− = − =
π
ππ ≈ π
π
≈ π π 2π
∴ π π
77
vii) Trigonometric Identities
θ θ θ θ+ =
θ θ θ
θ θ= − θ θ= −
θ
θ
θθ=
θθ
=
θθ
=
θθ
=
θ θθ
=
θ
78
=
÷
∴
=
÷
∴
−
∴
79
−−
+=
( ) ( )( )( ) + − −
− +
÷
−
∴
π
= − =
− = ∴ ∴ − − π π
viii) Graphs of y = sin x and y = cos x
π
π
π π π π
π π π
80
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
81
A C A D E M I C S K I L L S C E N T R E ( A S C )
The Academic Skills Centre supports and empowers undergraduate and graduate students by providing flexible instruction in the skills necessary for them to succeed at university: the ability to think critically, communicate their ideas effectively, and take responsibility for their own learning.
Phone (705) 748-1720 Fax (705) 748-1830 e-mail: [email protected]/academicskills
