Analytic Trigonometry

1
Analytic Trigonometry

• Chapter 6

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2
The Inverse Sine, Cosine, and Tangent Functions

• Section 6.1

3
One-to-One Functions

• A one-to-one function is a function f such that
  any two different inputs give two different
  outputs
• Satisfies the horizontal line test
• Functions may be made one-to-one by restricting
  the domain

4
Inverse Functions

• Inverse Function Function f 1 which undoes the
  operation of a one-to-one function f.

5
Inverse Functions

• For every x in the domain of f,
• f 1(f(x)) x
• and for every x in the domain of f 1,
• f(f 1(x)) x
• Domain of f range of f 1, and range of f
  domain of f 1
• Graphs of f and f 1, are symmetric with respect
  to the line y x
• If y f(x) has an inverse, it can be found by
  solving x f(y) for y. Solution is y f 1(x)
• More information in Section 4.2

6
Inverse Sine Function

• The sine function is not one-to-one
• We restrict to domain

7
Inverse Sine Function

• Inverse sine function Inverse of the
  domain-restricted sine function

8
Inverse Sine Function

• y sin1x means x sin y
• Must have 1 x 1 and
• Many books write y arcsin x
• WARNING!
• The 1 is not an exponent, but an indication of
  an inverse function
• Domain is 1 x 1
• Range is

9
Exact Values of the Inverse Sine Function

• Example. Find the exact values of
• (a) Problem
• Answer
• (b) Problem
• Answer

10
Approximate Values of the Inverse Sine Function

• Example. Find approximate values of the
  following. Express the answer in radians rounded
  to two decimal places.
• (a) Problem
• Answer
• (b) Problem
• Answer

11
Inverse Cosine Function

• Cosine is also not one-to-one
• We restrict to domain 0, ¼

12
Inverse Cosine Function

• Inverse cosine function Inverse of the
  domain-restricted cosine function

13
Inverse Cosine Function

• y cos1x means x cos y
• Must have 1 x 1 and 0 y ¼
• Can also write y arccos x
• Domain is 1 x 1
• Range is 0 y ¼

14
Exact Values of the Inverse Cosine Function

• Example. Find the exact values of
• (a) Problem
• Answer
• (b) Problem
• Answer
• (c) Problem
• Answer

15
Approximate Values of the Inverse Cosine Function

• Example. Find approximate values of the
  following. Express the answer in radians rounded
  to two decimal places.
• (a) Problem
• Answer
• (b) Problem
• Answer

16
Inverse Tangent Function

• Tangent is not one-to-one (Surprise!)
• We restrict to domain

17
Inverse Tangent Function

• Inverse tangent function Inverse of the
  domain-restricted tangent function

18
Inverse Tangent Function

• y tan1x means x tan y
• Have 1 x 1 and
• Also write y arctan x
• Domain is all real numbers
• Range is

19
Exact Values of the Inverse Tangent Function

• Example. Find the exact values of
• (a) Problem
• Answer
• (b) Problem
• Answer

20
The Inverse Trigonometric Functions Continued

• Section 6.2

21
Exact Values Involving Inverse Trigonometric
Functions

• Example. Find the exact values of the following
  expressions
• (a) Problem
• Answer
• (b) Problem
• Answer

22
Exact Values Involving Inverse Trigonometric
Functions

• Example. Find the exact values of the following
  expressions
• (c) Problem
• Answer
• (d) Problem
• Answer

23
Inverse Secant, Cosecant and Cotangent Functions

• Inverse Secant Function
• y sec1x means x sec y
• j x j 1, 0 y ¼,

24
Inverse Secant, Cosecant and Cotangent Functions

• Inverse Cosecant Function
• y csc1x means x csc y
• j x j 1, y ? 0

25
Inverse Secant, Cosecant and Cotangent Functions

• Inverse Cotangent Function
• y cot1x means x cot y
• 1 lt x lt 1, 0 lt y lt ¼

26
Inverse Secant, Cosecant and Cotangent Functions

• Example. Find the exact values of the following
  expressions
• (a) Problem
• Answer
• (b) Problem
• Answer

27
Approximate Values of Inverse Trigonometric
Functions

• Example. Find approximate values of the
  following. Express the answer in radians rounded
  to two decimal places.
• (a) Problem
• Answer
• (b) Problem
• Answer

28
Key Points

• Exact Values Involving Inverse Trigonometric
  Functions
• Inverse Secant, Cosecant and Cotangent Functions
• Approximate Values of Inverse Trigonometric
  Functions

29
Trigonometric Identities

• Section 6.3

30
Identities

• Two functions f and g are identically equal
  provided f(x) g(x) for all x for which both
  functions are defined
• The equation above f(x) g(x) is called an
  identity
• Conditional equation An equation which is not an
  identity

31
Fundamental Trigonometric Identities

• Quotient Identities
• Reciprocal Identities
• Pythagorean Identities
• Even-Odd Identities

32
Simplifying Using Identities

• Example. Simplify the following expressions.
• (a) Problem cot µ tan µ
• Answer
• (b) Problem
• Answer

33
Establishing Identities

• Example. Establish the following identities
• (a) Problem
• (b) Problem

34
Guidelines for Establishing Identities

• Usually start with side containing more
  complicated expression
• Rewrite sum or difference of quotients in terms
  of a single quotient (common denominator)
• Think about rewriting one side in terms of sines
  and cosines
• Keep your goal in mind manipulate one side to
  look like the other

35
Key Points

• Identities
• Fundamental Trigonometric Identities
• Simplifying Using Identities
• Establishing Identities
• Guidelines for Establishing Identities

36
Sum and Difference Formulas

• Section 6.4

37
Sum and Difference Formulas for Cosines

• Theorem. Sum and Difference Formulas for
  Cosines
• cos( ) cos cos sin sin
• cos( ) cos cos sin sin

38
Sum and Difference Formulas for Cosines

• Example. Find the exact values
• (a) Problem cos(105)
• Answer
• (b) Problem
• Answer

39
Identities Using Sum and Difference Formulas
40
Sum and Difference Formulas for Sines

• Theorem. Sum and Difference Formulas for Sines
• sin( ) sin cos cos sin
• sin( ) sin cos cos sin

41
Sum and Difference Formulas for Sines

• Example. Find the exact values
• (a) Problem
• Answer
• (b) Problem sin 20 cos 80 cos 20 sin 80
• Answer

42
Sum and Difference Formulas for Sines

• Example. If it is known that
  and that find the
  exact values of
• (a) Problem cos(µ Á)
• Answer
• (b) Problem sin(µ Á)
• Answer

43
Sum and Difference Formulas for Tangents

• Theorem. Sum and Difference Formulas for
  Tangents

44
Sum and Difference Formulas With Inverse Functions

• Example. Find the exact value of each expression
• (a) Problem
• Answer
• (b) Problem
• Answer

45
Sum and Difference Formulas With Inverse Functions

• Example. Write the trigonometric expression as an
  algebraic expression containing u and v.
• Problem
• Answer

46
Key Points

• Sum and Difference Formulas for Cosines
• Identities Using Sum and Difference Formulas
• Sum and Difference Formulas for Sines
• Sum and Difference Formulas for Tangents
• Sum and Difference Formulas With Inverse Functions

47
Double-angle and Half-angle Formulas

• Section 6.5

48
Double-angle Formulas

• Theorem. Double-angle Formulas
• sin(2µ) 2sinµ cosµ
• cos(2µ) cos2µ sin2µ
• cos(2µ) 1 2sin2µ
• cos(2µ) 2cos2µ 1

49
Double-angle Formulas

• Example. If , find
  the exact values.
• (a) Problem sin(2µ)
• Answer
• (b) Problem cos(2µ)
• Answer

50
Identities using Double-angle Formulas

• Double-angle Formula for Tangent
• Formulas for Squares

51
Identities using Double-angle Formulas

• Example. An oscilloscope often displays a
  sawtooth curve. This curve can be approximated by
  sinusoidal curves of varying periods and
  amplitudes. A first approximation to the sawtooth
  curve is given by
• Show that
• y sin(2¼x)cos2(¼x)

52
Identities using Double-angle Formulas
53
Half-angle Formulas

• Theorem. Half-angle Formulas
• where the or sign is determined by the
  quadrant of the angle

54
Half-angle Formulas

• Example. Use a half-angle formula to find the
  exact value of
• (a) Problem sin 15
• Answer
• (b) Problem
• Answer

55
Half-angle Formulas

• Example. If , find the
  exact values.
• (a) Problem
• Answer
• (b) Problem
• Answer

56
Half-angle Formulas

• Alternate Half-angle Formulas for Tangent

57
Key Points

• Double-angle Formulas
• Identities using Double-angle Formulas
• Half-angle Formulas

58
Product-to-Sum and Sum-to-Product Formulas

• Section 6.6

59
Product-to-Sum Formulas

• Theorem. Product-to-Sum Formulas

60
Product-to-Sum Formulas

• Example. Express each of the following products
  as a sum containing only sines or cosines
• (a) Problem cos(4µ)cos(2µ)
• Answer
• (b) Problem sin(3µ)sin(5µ)
• Answer
• (c) Problem sin(4µ)cos(6µ)
• Answer

61
Sum-to-Product Formulas

• Theorem. Sum-to-Product Formulas

62
Sum-to-Product Formulas

• Example. Express each sum or difference as a
  product of sines and/or cosines
• (a) Problem sin(4µ) sin(2µ)
• Answer
• (b) Problem cos(5µ) cos(3µ)
• Answer

63
Key Points

• Product-to-Sum Formulas
• Sum-to-Product Formulas

64
Trigonometric Equations (I)

• Section 6.7

65
Trigonometric Equations

• Trigonometric Equations Equations involving
  trigonometric functions that are satisfied by
  only some or no values of the variable
• Values satisfying the equation are the solutions
  of the equation
• IMPORTANT!
• Identities are different
• Every value in the domain satisfies an identity

66
Checking Solutions of Trigonometric Equations

• Example. Determine whether the following are
  solutions of the equation
• (a) Problem
• Answer
• (b) Problem
• Answer

67
Solving Trigonometric Equations

• Example. Solve the equations. Give a general
  formula for all the solutions.
• (a) Problem
• Answer
• (b) Problem
• Answer

68
Solving Trigonometric Equations

• Example. Solve the equations on the interval 0
  x lt 2¼.
• (a) Problem
• Answer
• (b) Problem
• Answer

69
Approximating Solutions to Trigonometric Equations

• Example. Use a calculator to solve the equations
  on the interval 0 x lt 2¼. Express answers in
  radians, rounded to two decimal places.
• (a) Problem tan µ 4.2
• Answer
• (b) Problem 2 csc µ 5
• Answer

70
Key Points

• Trigonometric Equations
• Checking Solutions of Trigonometric Equations
• Solving Trigonometric Equations
• Approximating Solutions to Trigonometric Equations

71
Trigonometric Equations (II)

• Section 6.8

72
Solving Trigonometric Equations Quadratic in Form

• Example. Solve the equations on the interval 0
  x lt 2¼.
• (a) Problem
• Answer
• (b) Problem
• Answer

73
Solving Trigonometric Equations Using Identities

• Example. Solve the equations on the interval 0
  x lt 2¼.
• (a) Problem
• Answer
• (b) Problem
• Answer

74
Trigonometric Equations Linear in Sine and Cosine

• Example. Solve the equations on the interval 0
  x lt 2¼.
• (a) Problem
• Answer
• (b) Problem
• Answer

75
Trigonometric Equations Using a Graphing Utility

• Example.
• Problem Use a calculator to solve the equation
• 2 13sin x 14cos2 x
• on the interval 0 x lt 2¼. Express answers in
  degrees, rounded to one decimal place.
• Answer

76
Trigonometric Equations Using a Graphing Utility

• Example.
• Problem Use a calculator to solve the equation
• 2x 3cos x 0
• on the interval 0 x lt 2¼. Express answers in
  radians, rounded to two decimal places.
• Answer

77
Key Points

• Solving Trigonometric Equations Quadratic in Form
• Solving Trigonometric Equations Using Identities
• Trigonometric Equations Linear in Sine and Cosine
• Trigonometric Equations Using a Graphing Utility

78
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