Formula for Success a Mathematics Resource

Transcript

1 A C A D E M I C S K I L L S C E N T R E ( A S C ) Formula for Success a Mathematics Resource P e t e r b o r o u g h O s h a w a

2 Contents Section 1: Formulas and Quick Reference Guide 1. Formulas From Geometry 1 2. Interval Notation 3 3. Coordinate Geometry 4 4. Steps in completing the Square 5 5. The Graphs of the Eight Basic Functions 6 6. Radian Measure 8 7. Special Angles 9 8. The Unit Circle Inverse Trig Functions Trigonometric Identities Sine Law Cosine Law Graphs of Trig Functions 15 Section 2: Mathematics Review Modules 1. Order of Operations and Substitution Fractions, Decimals and Percent Exponents, Roots and Factorials 27 For information on this or any of our services, contact the Academic Skills Centre, Trent University, Peterborough, Ontario K9J 7B8 Phone (705) Fax (705) acdskills@trentu.ca Section 3: Calculus/Algebra Prep 1. Algebra i) Terminology 32 ii) Expanding and Factoring 33 iii) Factor Theorem and Synthetic Division Equations i) Definitions 40 ii) Linear Equations 41 iii) X and Y intercepts 42 iv) Solving Quadratics Using Factoring 43 v) Solving Quadratics Using Quadratic Formula 44 vi) Solving Absolute Value Equations 46 vii) Solving Polynomial Equations Exponents i) Exponents 49 ii) The Exponential Function and its Inverse the Logarithmic Function 51 iii) The Common Logarithm 53 iv) The Natural Logarithm 53 v) Laws of Logarithms 54 vi) Solving Equations with Exponents or Logarithms 56 vii) Expoential Growth and Decay Functions and Transformations i) Definition of a function 60 ii) Function Notation Inequalities i) Inequalities Definition 62 ii) Linear Inequalities 62 iii) Polynomial Inequalities 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 The Academic Skills Centre would like to thank the following for their contributions to this publication: Ruth Brandow, Ellen Dempsey, Marj Tellis, Lisa Davies.

3 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 c 2 = a 2 + b 2 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

4 Circle Formulae For a circle whose radius is r and diameter is d (d = 2r): Circumference : C = 2pr= pd Area: A = pr 2 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 = 4pr 2 Volume = 4/3 pr 3 Right Circular Cylinder A right circular cylinder where r is the radius of the base and h is the altitude has: Surface area = 2prh Right Circular Cone Volume = pr 2 h A right circular cone where r is the radius of the base and h is the altitude has: Surface area = p r 2 + h 2 2 Volume = 1/3 pr 2 h

5 2. Interval Notation { } { < } { < } 3

6 4 3. Co-ordinate Geometry = =

7 5 4. Steps in Completing the Square + + = = = + + = = = + + = = + + = + = + = + = + + = +

8 5. The Graphs of the Eight Basic Functions 6

9 7

10 6. Trigonometry - Radian Measure πππ π ππ π π π π π = = π π 8

11 7. Special Angles Finding The Exact Value of a Trigonometric Ratio 9

12 8. The Unit Circle 10

13 9. Inverse Trigonometric Functions (a.k.a. arcsin, arccos and arctan) π π π π [ π ] π π π π π 11

14 10. Trigonometric Identities =

15 11. Sine Law 13

16 12. Cosine Law + 14

17 13. Graphs of Trigonometric Functions 15

18 16 Section 2 Mathematics Review Modules 1. Order of Operations and Substitution Order of Operations (4) = = = = = 16

19 + + = = / + = / 17

20 Substituting into Formulae = = = For Practice Problems on this section, please visit: 18

21 19

22 20 2. Fractions, Decimals, and Percents Includes Rounding and Significant Digits Fractions = =

23 + = + = = = = = = = = = = 21

24 + + = + + = + = + = + = + = 22

25 Percents 23

26 24 Decimals =

27 Rounding and Significant Digits 25

28 For Practice Problems on this section, please visit: 26

29 3. Powers, Roots and Factorials Powers 27

30 28

31 Roots = + = = = + + = = 29

32 Laws of Exponents = = = = = = ( ) = = = = = = = ( ) = = = = = = = = = = = = = = ( ) 30

33 Factorials = / / / / / / / / / / = = = = = For Practice Problems on this section, please visit: 31

34 Section 3 Calculus/Algebra Prep 1. Essential Algebra i) Terminology

35 + + + = ii) Expanding and Factoring

36 + + + ( ) 34

37 = + = = = + 35

38 = = ( + ) + ( ) ( ) 36

39 + + = + + = + + = =

40 iii) Factor Theorem (polynomial of degree three or greater) + + = = = = 38

41 For Practice Problems on this section, please visit: 39

42 2. Equations of Straight Lines i) Definitions + 40

43 Equations ii) Linear Equations + = + = ( ) + = + = ( ) 41

44 iii) X and Y intercepts 42

45 iv) Quadratic Equations Using Factoring + = = + = + + = = + + = 43

46 v) Solving Quadratics Using Quadratic Formula = ± + + = + + = ( ) + = ( ) + = ( ) + = ± + = = ± = 44 = ± = ± = ± = ± = ±

47 = = ± = ± = ± = = = 45

48 vi) Absolute Value Equations = = = = < = + = = = = < 46

49 vii) Solving Polynomial Equations (With Degree Three or More) + + = + + = + + = + = + + = = = ± ± = ± = 47

50 + = = + = + = + + = = = = + = = + + = = ± = ± + + = = + = = [ ] + = + = For Practice Problems on this section, please visit: 48

51 3. Exponents i) Laws of Exponents + = = = = = = = = = = ( ) = = ( ) = = = = = = = = = = = = = = = = = = 49

52 = = = = = = = = = = ( ) = = 6. ( ) ( ) = 50 = ( ) ( )

53 = ( ) ( ) = = ( ) ( ) = ( ) = = = ( ) = = ii) The Exponential Function and Its Inverse The Logarithmic = Function = = = = = = = = = = 51

54 = = = = = = = = = = = = = = = ( ) = = = = = = = = = 52

55 iii) The Common Logarithm = = = = iv) The Natural Logarithm (ln and e) = 53

56 v) Laws of Logarithms = + = + = ( ) = = = = = = = + = [ ] = = ( ) = = = = = 54

57 = = + = = = ( ) = = = = 55

58 vi) Solving Equations with Exponents or Logarithms = = + = + ( ) = = + = + = = ± = + + = + = = + + = 56

59 = = = = = = = = = = = = + = = = 57

60 vii) Exponential Growth and Decay = 58

61 = + = For Practice Problems on this section, please visit: 59

62 4. Functions and Transformations i) Definition of a Function 60

63 ii) Function Notation = = = = For Practice Problems on this section, please visit: 61

64 5. Inequalities i) Inequalities Definition ii) Linear Inequalities 62

65 ii) Polynomial Inequalities (Quadratic, Cubic and Quartic Inequalities) > > + + [ ] [ ] 63

66 < < < < = = (( ( )) ) ( ( ( )) ) (( ( )) ) + + > > (( ( )( )( )( )) > ) > = ± = ± = ± = ± (( ( )) ) (( ( )) ) ( ) ( ) ( ) (( )) (( )) (( )) 64

67 Polynomial Inequalities (With Non Real Roots) ( )( ) + + ( + + ) = ± -4 4 ( ] [ ] [ ) For Practice Problems on this section, please visit: 65

68 6. Trigonometric Functions i) Definitions + = = = = = = = 66

69 = = = = = = = = 67

70 ii) CAST Rule + = + = 68

71 + = + = ± = ± 69

72 iii) Radian Measure θ = θ π θ = π π = π π = = π π ( ) π ( ) π π π ( ) π π π π = π π = ( π ) ( ) π = = π 70

73 iv) Finding Exact Values Of Trig Ratios Special Angles And The Unit Circle π π ( ) ( ) π = = 71

74 π θ = ( ) = = = π π π π π = 72

75 π 6 π 4 π 3 π π = π = π/2 (0,1) π = π ( 1,0)0 (1,0) π = π π π = π π 2π π π π = = π π π π = = π π π π = π = = π 73

76 v) Using The Calculator To Find The Ratios 74 2π π π π π π π π = π π π π

77 π π = π π π π ( ) = = 2π π 1.27 π π π. 75

78 vi) Trigonometric Equations π 2π π π π π = π 2π = = = π π π π + = π = = π π ππ π π 2π π π 76

79 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

80 = = 78

81 = + ( + ) ( ) + ( )( ) π = = = π π 79

82 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: 80

83 81

84 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) Fax (705) acdskills@trentu.ca