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Analytic Trigonometry

Analytic Trigonometry. Chapter 6. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A. The Inverse Sine, Cosine, and Tangent Functions. Section 6.1. One-to-One Functions.

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Analytic Trigonometry

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

  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: 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{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: 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{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: 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{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: 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

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