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Section 5.5 Inverse Trigonometric Functions & Their Graphs

Chapter 5 – Trigonometric Functions: Unit Circle Approach. Section 5.5 Inverse Trigonometric Functions & Their Graphs. Review of Inverse Functions. Remember If the graph passes the horizontal line test, then the function has an inverse functions.

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Section 5.5 Inverse Trigonometric Functions & Their Graphs

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  1. Chapter 5 – Trigonometric Functions: Unit Circle Approach Section 5.5 Inverse Trigonometric Functions & Their Graphs 5.5 - Inverse Trigonometric Functions & Their Graphs

  2. Review of Inverse Functions Remember • If the graph passes the horizontal line test, then the function has an inverse functions. • If a point (a, b) is on the graph of f, then the point (b, a) is on the graph of f -1. • The graph of f -1 is a reflection of the graph of f about the line y=x. 5.5 - Inverse Trigonometric Functions & Their Graphs

  3. Sine Function • Does not pass the horizontal line test. • Must restrict the domain to create an inverse function. 5.5 - Inverse Trigonometric Functions & Their Graphs

  4. Definition • The inverse sine function is the function sin-1 with domain [-1, 1] and range [- ⁄ 2,  ⁄ 2] defined by • The inverse sine function is also called arcsine denoted by arcsin. 5.5 - Inverse Trigonometric Functions & Their Graphs

  5. Note 5.5 - Inverse Trigonometric Functions & Their Graphs

  6. Graph of Inverse sine 5.5 - Inverse Trigonometric Functions & Their Graphs

  7. Cancellation Properties - Sine • Thus y = sin-1x is the number in the interval [- ⁄ 2,  ⁄ 2] whose sine is x. • In other words we have the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  8. Examples Find the exact value of the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  9. Cosine Function • Does not pass the horizontal line test. • Must restrict the domain to create an inverse function. 5.5 - Inverse Trigonometric Functions & Their Graphs

  10. Definition • The inverse cosine function is the function cos-1 with domain [-1, 1] and range [0, ] defined by • The inverse sine function is also called arccosine denoted by arccos. 5.5 - Inverse Trigonometric Functions & Their Graphs

  11. Graph of Inverse Cosine 5.5 - Inverse Trigonometric Functions & Their Graphs

  12. Cancellation Properties - Cosine • Thus y = cos-1x is the number in the interval [0, ] whose cosine is x. • In other words we have the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  13. Examples Find the exact value of the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  14. Tangent Function • Does not pass the horizontal line test. • Must restrict the domain to create an inverse function. 5.5 - Inverse Trigonometric Functions & Their Graphs

  15. Definition • The inverse tangent function is the function tan-1 with domain (-∞, ∞) and range (- ⁄ 2,  ⁄ 2)defined by • The inverse tangent function is also called arctangent denoted by arctan. 5.5 - Inverse Trigonometric Functions & Their Graphs

  16. Graph of Inverse Tangent 5.5 - Inverse Trigonometric Functions & Their Graphs

  17. Cancellation Properties - Tangent • Thus y = tan-1x is the number in the interval (- ⁄ 2,  ⁄ 2)whose sine is x. • In other words we have the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  18. Examples Find the exact value of the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  19. Evaluating Compositions 5.5 - Inverse Trigonometric Functions & Their Graphs

  20. Inverse Properties 5.5 - Inverse Trigonometric Functions & Their Graphs

  21. Using Inverse Properties Evaluate the following: 5.5 - Inverse Trigonometric Functions & Their Graphs

  22. Examples – pg. 412 • Find the exact value of the expression if it is defined. 5.5 - Inverse Trigonometric Functions & Their Graphs

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