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 = sin{1x 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:

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:

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 = cos{1x 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:
(c) Problem:

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:

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 = tan{1x 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:

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:

22. Exact Values Involving Inverse Trigonometric Functions
Example. Find the exact values of the following expressions
(c) Problem:
Answer:
(d) Problem:

23. Inverse Secant, Cosecant and Cotangent Functions
Inverse Secant Function
y = sec{1x means x = sec y
j x j ¸ 1, 0 · y · ¼,

24. Inverse Secant, Cosecant and Cotangent Functions
Inverse Cosecant Function
y = csc{1x means x = csc y
j x j ¸ 1, y  0

25. Inverse Secant, Cosecant and Cotangent Functions
Inverse Cotangent Function
y = cot{1x means x = cot y
{1 < x < 1, 0 < y < ¼

26. Inverse Secant, Cosecant and Cotangent Functions
Example. Find the exact values of the following expressions
(a) Problem:
Answer:
(b) Problem:

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:

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:

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:

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±

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)
cos \theta = 12/13
cos \phi = -24/25

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:

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)

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)
Only use the double angle formula on the second summand!

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:

55. Half-angle Formulas Example. If , find the exact values. (a) Problem:
Answer:
(b) Problem:

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)
(c) Problem: sin(4µ)cos(6µ)

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)

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:

67. Solving Trigonometric Equations
Example. Solve the equations. Give a general formula for all the solutions.
(a) Problem:
Answer:
(b) Problem:

68. Solving Trigonometric Equations
Example. Solve the equations on the interval 0 · x < 2¼.
(a) Problem:
Answer:
(b) Problem:

69. Approximating Solutions to Trigonometric Equations
Example. Use a calculator to solve the equations on the interval 0 · x < 2¼. Express answers in radians, rounded to two decimal places.
(a) Problem: tan µ = 4.2
Answer:
(b) Problem: 2 csc µ = 5

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 < 2¼.
(a) Problem:
Answer:
(b) Problem:

73. Solving Trigonometric Equations Using Identities
Example. Solve the equations on the interval 0 · x < 2¼.
(a) Problem:
Answer:
(b) Problem:

74. Trigonometric Equations Linear in Sine and Cosine
Example. Solve the equations on the interval 0 · x < 2¼.
(a) Problem:
Answer:
(b) Problem:

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 < 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 < 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