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Presentation on theme: "Analytic Trigonometry Chapter 6 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AAA A A A A."— Presentation transcript:

1 Analytic Trigonometry Chapter 6 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AAA A A A A

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 {1 x 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 = cos {1 x 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 = tan {1 x 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 = sec {1 x means x = sec y j x j ¸ 1, 0 · y · ¼,

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

25 Inverse Secant, Cosecant and Cotangent Functions Inverse Cotangent Function y = cot {1 x 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: 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) = cos 2 µ { sin 2 µ cos(2µ) = 1 { 2sin 2 µ cos(2µ) = 2cos 2 µ { 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)cos 2 (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 < 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 < 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 < 2¼. (a) Problem: Answer: (b) Problem: Answer:

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

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

75 Trigonometric Equations Using a Graphing Utility Example. Problem: Use a calculator to solve the equation sin x = 14cos 2 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

78


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