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ACADEMIC SKILLS CENTRE (ASC)
Formula for Success
a Mathematics Resource
Peterborough • Oshawa
Contents
Section 1: Formulas and Quick
Reference Guide
1. Formulas From Geometry
2. Interval Notation
3. Coordinate Geometry
4. Steps in completing the Square
5. The Graphs of the Eight
Basic Functions
6. Radian Measure
7. Special Angles
8. The Unit Circle
9. Inverse Trig Functions
10. Trigonometric Identities
11. Sine Law
12. Cosine Law
13. Graphs of Trig Functions
Section 3: Calculus/Algebra Prep
1
3
4
5
6
8
9
10
11
12
13
14
15
Section 2: Mathematics Review
Modules
1. Order of Operations and
Substitution.
2. Fractions, Decimals and Percent
3. Exponents, Roots and Factorials
For information on this
or any of our services, contact
the Academic Skills Centre,
Trent University, Peterborough,
Ontario K9J 7B8
16
20
27
1.
i)
ii)
iii)
Algebra
Terminology
32
Expanding and Factoring
33
Factor Theorem and Synthetic Division 38
2.
i)
ii)
iii)
iv)
v)
40
41
42
43
Equations
Definitions
Linear Equations
X and Y intercepts
Solving Quadratics Using Factoring
Solving Quadratics Using Quadratic
Formula
vi) Solving Absolute Value Equations
vii) Solving Polynomial Equations
3. Exponents
i) Exponents
ii) The Exponential Function and its
Inverse the Logarithmic Function
iii) The Common Logarithm
iv) The Natural Logarithm
v) Laws of Logarithms
vi) Solving Equations with Exponents
or Logarithms
vii) Expoential Growth and Decay
4.
i)
ii)
5.
i)
ii)
iii)
44
46
47
49
51
53
53
54
56
58
Functions and Transformations
Definition of a function
Function Notation
60
61
Inequalities
Inequalities Definition
Linear Inequalities
Polynomial Inequalities
62
62
63
6. Trigonometric Functions
i) Trig Definitons
66
ii) CAST Rule
68
iii) Radian Measure
70
iv) Finding Exact Values of Trig Ratios 71
v) Using a calculator to find Trig Ratios 74
vi) Trig Equations
76
vii) Trig Identities
77
viii) Graphs of y = sinx and y = cosx
80
Phone (705) 748-1720
Fax (705) 748-1830
e-mail: acdskills@trentu.ca
www.trentu.ca/academicskills
The Academic Skills Centre would like to thank the
following for their contributions to this publication:
Ruth Brandow, Ellen Dempsey, Marj Tellis, Lisa Davies.
Section 1
Formulas and Quick Reference Guide
1. Formulas from Geometry
Pythagoream Theorem
If a triangle has sides a, b, and c
(c is the hypotenuse), then c2 = a2 + b2
Triangle with Base and Height Given
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
Triangle with Base and Height Unknown
The area of a triangle with angles A, B, and C and
sides opposite a, b, and c, respectively:
Area = ½ ab x sinC
Area of a Parallelogram
The area of a parallelogram with base b
and altitude h is Area = bh
Area of a Trapezoid
The area of a trapezoid whose parallel sides are a
and b and altitude is h is
Area = ½(a + b)h
1
Circle Formulae
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
3 Dimensional Solids
Sphere
A Sphere with radius r has:
Surface Area = 4pr2
Volume = 4/3 pr3
Right Circular Cylinder
A right circular cylinder where r is the radius
of the base and h is the altitude has:
Surface area = 2prh
Volume = pr2h
Right Circular Cone
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
2
2. Interval Notation
{ ≤ ≤ }
{ ≤ < }
{ < ≤ }
∞
∞
∞
≥
∞
≥
∞ ∞
3
3. Co-ordinate Geometry
=
=
−
−
− + −
+
+ +
−
4
4. Steps in Completing the Square
+ + = + +
+
+ = + + − +
−
−
=
+
+
+ + = +
+
+ + = + + − + = + +
− + = − + = − + − − − + = − − = − −
− −
+ = − − + − −
+ = − − + = − − −
5
5. The Graphs
of the Eight Basic Functions
6
7
6. Trigonometry - Radian Measure
πππ
π
ππ
π
π
π
π
=
π
=
π
π
8
7. Special Angles
Finding
The Exact Value of a Trigonometric Ratio
9
8. The Unit Circle
10
9. Inverse Trigonometric Functions
(a.k.a. arcsin,
arccos
and arctan)
π π
−
− ≤ ≤ −
π
≤≤
π
[ π ]
− ≤ ≤ ≤ ≤ π
π
π
π
π
− −
11
10. Trigonometric
Identities
=
+
−
−
+
−
12
11. Sine Law
13
12.
Cosine Law
+ −
14
13. Graphs of Trigonometric Functions
15
Section 2
Mathematics Review Modules
1.
Order of Operations and Substitution
Order of Operations
+ − − ÷ +
10
12÷÷+3+ 2
++2(4)−−
==
10++8− −
12
÷ ÷+3+ 2
==
10++8− −+4+ 2
==
18+
−+4+ 2
==
14+ + 2
== 16
16
+
+
= =
/ +
=
/
17
Substituting
into Formulae
+
+
+
=
=
=
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
18
19
2. Fractions, Decimals, and Percents
Includes
Rounding and
Significant Digits
Fractions
•
•
•
÷= =
•
•
•
20
=
=
×
=
×
=
=
÷ = =
+
+ =
÷
= ×
= =
21
+
+ =
+
+ =
+
=
+
+ =
+
=
=
22
Percents
23
Decimals
÷=
÷
÷
24
Rounding
and Significant
Digits
•
•
25
•
•
•
•
•
•
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
26
3.
Powers, Roots and Factorials
Powers
27
28
=
+ = =
=
Roots
+ ≠ +
=
=
29
Laws of Exponents
⇒ ⋅ =
÷ = − ⇒ ÷ =−
( ) =
= ⇒(
) =
=
= ⇒( −) =
⇒− =−
⇒− = =
= ⇒ = =
= ⇒ = = =
−
−
⇒ = = =
30
−
⇒ = ⋅ =
−
= = ⇒ =
=
Factorials
/ / / / /
=
/ / / / /
=
=
=
⋅
=
=
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
31
Section 3 Calculus/Algebra Prep
1. Essential Algebra
i) Terminology
− +
−
− +
−
32
− +
≠
− +
+ + =
ii) Expanding and Factoring
−
+ − −
+ −
33
+ −
− +
( − + )
34
−
− = − +
=
+ − +
+ + + +
− −
∴ − − = − +
+ −
∴ + − = + −
35
+
+
+ +
≠ ≠ ≠ ≠
+ + + +
+ −−
+
∴
∴
+ −− ==
− −−
+
+ +−
+ +
( + )
− +
( − )
36
( )
+ −
+ = + − + − = − + +
−
= − + +
+
= + − +
37
iii) Factor Theorem
(polynomial of degree
three or greater)
+ − + ÷ −
−
= ∴ =
∴
+ +
+ − + = − + + +
− + −
∴ − − − + − − =
38
− −−−
−
−
+ +
∴ ∴ − −−
−
− −÷ ÷ + +
− −
−
− − +
+
− − + ++ ++ +
− − + +
+
− − + + +
∴
∴
+ +−−++
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
39
2. Equations of Straight
Lines
i) Definitions
−
−
− + −
40
Equations
ii) Linear Equations
∴
− +
∴
−
+
−
= ()
−
=
−
=
+
−
−
= ()
−
+
41
iii) X and Y intercepts
∴
≠
∴
42
iv) Quadratic Equations Using Factoring
− + =
∴
∴
= −
+ =
∴
∴
+ − += + − +
− − + = + − +
− − =
∴
∴
−
43
v) Solving Quadratics Using
Quadratic
Formula
=
− ± −
+ + =
+
+
−
= −
( +
)
−
= −
( +
)
=
−
( + ) =
−
+
= ±
−
=
− ± −
− − =
=
44
± − −
∴ =
±
∴ =
±
∴ =
±
∴ =
±
− = −
=
± −
∴ =
± −
−
∴ =
±
− − =
− = − − =
45
vi) Absolute Value Equations
=
= −
= ⇒ ≥ = < = −
+ =
⇒ ∴
⇒ ∴
− = −
⇒ ∴ =
≥
⇒ ⇒ ∴ =
<
46
−
vii) Solving Polynomial Equations
(With Degree Three or More)
+ + =
+ + =
+ + =
+ =
+ − + =
±
= − =
−
=
± −
=
±
47
− + =
= − +
∴
= − + −
∴ = − + −
∴ − + − =
∴ = = − =
− + =
− − =
− + − + =
∴
∴ = ± = ±
∴ ∴
∴ ∴
∴ − + + = ∴ − − + − + − − + − =
∴ − − + − = = −
[
]
∴ − − + − − =
∴ − + − =
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
48
3. Exponents
i) Laws of Exponents
≠
= +
⇒ ⋅ =
÷ = −
⇒ ÷ = −
= ⇒ = ⋅ =
=
()
=
⇒ = × =
= − ⇒
( )
=
=
⇒ − =
⇒ − = −
− =
=
=
⇒ − =
=
⇒ = =
⇒ = = =
−
⇒ =
−
=
=
−
=
−
49
−
=
−
−
− −
=
− −
−
− =
=
− =
−
−
−
=
−
=
−
=
−
=
−
=
( )
=
=
6.
(
−
)
−
(
)
−
−
=
− −
=
( )
50 ( )
−
−
=
( )
=( )
−
=
−
(
−
−
−
=
=
=
)
)
−
(
( )
=
=
−
−
−
( )
−
−
−
=
=
ii) The Exponential Function and Its
Inverse
The Logarithmic Function
=
=
=
=
=
= ⇒ =
=
=
=
= ⇒ =
51
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
= −
()
−
=
=
=
=
=
−
=
= −
=
=
52
iii)
The Common Logarithm
=
=
− =
=
iv)
The Natural Logarithm
(ln and e)
=
53
v) Laws of Logarithms
= + ⇒ = +
= −
=
=
=
⇒
( ) = −
⇒ =
⇒ =
⇒ =
+
[
=
]
=
−
= (
)
=
=
=
=
=
54
=
=
= +
=
=
=
( )
= −
=
≈
=
≈
=
55
vi) Solving Equations with Exponents or
Logarithms
=
=
≈
+ − − =
( +− ) =
=
+
−
= +
=
=
±
= ≠ −
+ + =
+ =
= +
+ − =
∴
56
=
=
− =
− =
− =
− =
≈
− =
− − =
= = −
− − =
− − =
− + =
= = −
∴
57
vii) Exponential Growth and Decay
=
÷
≈
∴
58
= +
≈
∴
=
≈
∴
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
59
4. Functions and Transformations
i) Definition of a Function
60
ii) Function Notation
=
=
−
−
−
− −
= −
=
− −
−
≠
≠
∴
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
61
5. Inequalities
i) Inequalities Definition
≥ ≤
ii) Linear Inequalities
≤
62
− ∞
ii) Polynomial Inequalities
(Quadratic,
Cubic and Quartic Inequalities)
− >
− − >
∴
− ∞ − ∞
− ∞ − ∞
− ≤ + +
− − ≤
−
−
− ∞ −
[
]
[
] ∞
−
∴ −
−
−
63
⇒
⇒
⇒
−−∞∞
∞∞
∞
−−∞
−
−
∴
∴
∴
∞∞
∞
−
−− <<<
−−−===
⇒
⇒
⇒
− ∞
((−−∞
(∞
))
)
((∞(∞))∞ )
−−−<<<
− ∞
∴
∴
∴
≠≠
≠
((−−∞
(∞
)) )
−−
−
+++>>>
(( ( )()( )( )) )
−−− −−− >>>
==±±=±
==±±=±
⇒
⇒⇒
((−−∞(∞−−∞−)−)
)
((∞(∞))∞)
∴
∴
∴
((−−∞(∞−−∞−)−)(()−− (−))(()∞(∞))∞ )
64
Polynomial Inequalities
(With Non
Real Roots)
(
)(
)
− + + ≤
(
)
+ + ≤
−
⇒ = ±
( − ∞ −] [ − ] [ ∞ )
4
-4
∴
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
65
6. Trigonometric Functions
i) Definitions
°
°
°
−°
°
+ =
=
66
=
=
=
=
=
=
=
=
=
=
=
=
=
°
°
°
°
67
ii) CAST Rule
°
° ° − °
°
+ =
+ − =
68
−
−
∴
+ =
+ =
±
∴ = ±
69
iii) Radian Measure
θ = θ
°
π
π ° π
∴ ° = π
°
π
=
∴ ° =
π
°
θ =
°
°
( π )
π
(
)
− °
π
− π
π
π
−π
π
( π ) = °
(
( π ) =
= °
π
−π
70
π
) = −°
π
iv) Finding Exact Values Of Trig Ratios
Special Angles
And The Unit Circle
π (°) π (°) π °
=
=
71
θ =
π
°
(
)
=
=
×
=
π
π
π
π ∴ π =
−
72
π
3
π
4
π
6
π = − π
−
π
=
π/2 (0,1)
∴ π =
π (−1,0)0 (1,0)
∴ π =
π
π − π =
−
2π π π
π
π
=
π
π
=
−
−π = π =
π = − π
= −
π
π − π = π
π
73
v) Using The Calculator To Find The Ratios
−
°
°
° −
2π
−
π
π π π π
π =
π
π π
74
π π
− − −
−
π
=
π
π π
π π
− − −
°
− () = ° − °
°
≈ ° °
−
−
− = °
°
°
2π
−
−
≈ π − 1.27 π + 1.27 π
≈
π.
75
vi) Trigonometric Equations
≤ ≤π
2π
−
π π
π =
π π
2π
− − =
− − =
∴ = −
π
π
π
π
+ − =
∴
−
− =
− = π
≈ ππ
π π
≈
π
π
2π
∴ π π
76
vii) Trigonometric Identities
θ θ
θ
θ
θ
θ + θ =
θ = − θ
θ = − θ
θ
θ
θ =
θ
θ =
θ
θ =
θ
θ =
θ
θ =
θ
θ
θ
77
=
÷
∴
=
÷
∴
−
∴
78
−
=
− +
( + ) − ( − )
( − )( + )
÷
−
∴
π
= −
=
− =
∴
∴ − −
π
π
79
viii) Graphs of y = sin x and y = cos x
π
π π π π π π π π
For Practice Problems on this section, please visit:
www.trentu.ca/academicskills/online_math.php
80
81
ACADEMIC SKILLS CENTRE (ASC)
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: acdskills@trentu.ca
www.trentu.ca/academicskills
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