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Pre-Calculus

Pre-Calculus. Chapter 4 Trigonometric Functions. 4.7 Inverse Trigonometric Functions. Objectives: Evaluate inverse sine functions. Evaluate other inverse trigonometric functions. Evaluate compositions of trigonometric functions. Inverse Functions.

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Pre-Calculus

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  1. Pre-Calculus Chapter 4 Trigonometric Functions

  2. 4.7 Inverse Trigonometric Functions Objectives: • Evaluate inverse sine functions. • Evaluate other inverse trigonometric functions. • Evaluate compositions of trigonometric functions.

  3. Inverse Functions • Recall that a function and its inverse reflect over the line y = x. • What must be true for a function to have an inverse? • It must be one-to-one, that is, it must pass the horizontal line test.

  4. More Inverse Functions • Are sine, cosine, and tangent one-to-one? • If not, what must we do so that these functions will have inverse functions? • Hint: Consider y = x2. • We must restrict the domain of the original function.

  5. y x y = sin x Sine and Its Inverse • f(x) = sin xdoes not pass the Horizontal Line Test • It must be restricted to find its inverse. Sin x has an inverse function on this interval.

  6. Angle whose sine is x Inverse Sine Function • The inverse sine function is defined by y = arcsinxif and only if sin y = x. • The domain of y = arcsinx is [–1, 1]. • The range of y = arcsinxis _____________. • Why are the domain and range defined this way?

  7. What Does “arcsin” Mean? • In an inverse function, the x-values and the y-values are switched. • So, arcsinx means the angle (or arc) whose sin is x. • Notation for inverse sine • arcsinx • sin -1x

  8. Examples • If possible, find the exact value.

  9. Graphing Arcsine • Create a table for sin y = xfor –π/2 ≤ y ≤ π/2. • Graph x on horizontal axis and y on vertical axis.

  10. Graph of Arcsine

  11. y x y = cos x Inverse Cosine Function • f(x) = cosx must be restricted to find its inverse. Cos x has an inverse function on this interval.

  12. Angle whose cosine is x Inverse Cosine Function • The inverse cosine function is defined by y = arccosxif and only if cosy= x. • The domain of y = arccosx is [–1, 1]. • The range of y = arccosxis [0, π]. • Notation for inverse cosine: • arccosx orcos-1x

  13. Examples • If possible, find the exact value

  14. Graphing Arccos • Create a table for cosy = xfor 0 ≤ y ≤ π. • Graph x on horizontal axis and y on vertical axis.

  15. Graph of Arccos

  16. y = tan x y x Inverse Tangent Function • f(x) = tan x must be restricted to find its inverse. Tan x has an inverse function on this interval.

  17. Angle whose tangent is x Inverse Tangent Function • The inverse tangent functionis defined by y = arctanxif and only if tan y= x. • The domain of y = arctanx is (–∞, ∞). • The range of y = arctanxis (–π/2, π/2). • Notation for inverse tangent: • arctanx ortan -1x

  18. Examples • If possible, find the exact value

  19. Graph of Arctan

  20. Examples • Evaluate using your calculator. (What mode should the calculator be in?)

  21. Summary

  22. Composition of Functions • Given the restrictions specified in the previous slide, we have the following properties of inverse trig functions.

  23. Examples • If possible, find the exact value.

  24. y x u Example 3 2

  25. Example • Find the exact value of .

  26. Homework 4.7 • Worksheet 4.7

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