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4.7 Inverse Trig Functions

4.7 Inverse Trig Functions. Inverse trig functions. What trig functions can we evaluate without using a calculator? Sin Cos Tan Sin. Inverse Trig Functions . What does an inverse function do? Finds the input of a function when given the output

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4.7 Inverse Trig Functions

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  1. 4.7 Inverse Trig Functions

  2. Inverse trig functions • What trig functions can we evaluate without using a calculator? • Sin • Cos • Tan • Sin

  3. Inverse Trig Functions • What does an inverse function do? • Finds the input of a function when given the output • How can we determine if a function has an inverse? • Horizontal Line Test • If any horizontal line intersects the graph of a function in more than one point, the function does not have an inverse

  4. Does the Sine function have an inverse? 1 -1

  5. What could we restrict the domain to so that the sine function does have an inverse? 1 -1

  6. Inverse Sine, , arcsine (x) • Function is increasing • Takes on full range of values • Function is 1-1 • Domain: • Range:

  7. Evaluate: arcSin • Asking the sine of what angle is

  8. Find the following: • ArcSin • ArcSin

  9. Inverse Cosine Function • What can we restrict the domain of the cosine curve to so that it is 1-1? 1 -1

  10. Inverse Cosine, , arcCos (x) • Function is increasing • Takes on full range of values • Function is 1-1 • Domain: • Range:

  11. Evaluate: ArcCos (-1) • The Cosine of what angle is -1?

  12. Evaluate the following: • ArcCos

  13. arcCos (0.28) • Is the value 0.28 on either triangle or curve? • Use your calculator:

  14. Determine the missing Coordinate

  15. Determine the missing Coordinate

  16. Use an inverse trig function to write θ as a function of x. 2x θ x + 3

  17. Find the exact value of the expression. Sin [ ArcCos ]

  18. 4.7 Inverse Trig Functions

  19. So far we have: • Restricted the domain of trig functions to find their inverse • Evaluated inverse trig functions for exact values

  20. ArcTan (x) • Similar to the ArcSin (x) • Domain of Tan Function: • Range of Tan Function:

  21. Composition of Functions From Algebra II: If two functions, f(x) and (x), are inverses, then their compositions are: f((x)) = x and (f(x)) = x

  22. Inverse Properties of Trig Functions • If -1 ≤ x ≤ 1 and - ≤ y ≤ , then Sin (arcSin x) = x and arcSin (Sin y) = y • If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π, then Cos (arcCosx) = x and arcCos (Cos y) = y • If x is a real number and - < y <, then Tan (arcTanx) = x and arcTan (Tan y) = y

  23. Inverse Trig Functions • Use the properties to evaluate the following expression: Sin (ArcSin 0.3)

  24. Inverse Trig Functions • Use the properties to evaluate the following expression: ArcCos (Cos )

  25. Inverse Trig Functions • Use the properties to evaluate the following expression: ArcSin (Sin 3π)

  26. Inverse Trig Functions • Use the properties to evaluate the following expression: • Tan (ArcTan 25) • Cos (ArcCos -0.2) • ArcCos (Cos )

  27. 4.7 Inverse Trig Functions

  28. Inverse Trig Functions • Yesterday, we only had compositions of functions that were inverses • When we have a composition of two functions that are not inverses, we cannot use the properties • In these cases, we will draw a triangle

  29. Inverse Trig Functions • Sin (arcTan ) • Let u = whatever is in parentheses • u = arcTan → Tan u =

  30. Inverse Trig Functions • Sec (arcSin )

  31. Inverse Trig Functions • Sec (arcSin ) • Cot (arcTan - ) • Sin (arcTan x)

  32. Inverse Trig Functions • In this section, we have: • Defined our inverse trig functions for specific domains and ranges • Evaluated inverse trig functions • Evaluated compositions of trig functions • 2 Functions that are inverses • 2 Functions that are not inverses by evaluating the inner most function first • 2 Functions that are not inverses by drawing a triangle

  33. Sine Function 1 - -1

  34. Cosine Function 1 π -1

  35. Tangent Function -

  36. Evaluating Inverse Trig Functions • arcTan (- ) • ) • arcSin (-1)

  37. Composition of Functions • When the two functions are inverses: • Sin (arcSin -0.35) • arcCos (Cos )

  38. Composition of Functions • When the two functions are not inverses: • (Cos ) • arcTan (Sin )

  39. Composition of Functions • When the two functions are not inverses: • Sin (arcCos ) • Cot ( )

  40. 4.7 Inverse Trig Functions

  41. Inverse Trig Functions • Evaluate the following function: f(x) = Sin (arcTan 2x) In your graphing calculator, graph both of these functions.

  42. Inverse Trig Functions • Solve the following equation for the missing piece: arcTan = arcSin(___)

  43. Inverse Trig Functions • Find the missing pieces in the following equations: • arcSin = arcCos (___) • arcCos = arcSin (___) • arcCos = arcTan (___)

  44. Inverse Trig Functions

  45. Inverse Trig Functions

  46. Composition of Functions • Evaluate innermost function first • Substitute in that value • Evaluate outermost function

  47. Sin (arcCos ) Evaluate the innermost function first: arcCos ½ = Substitute that value in original problem

  48. How do we evaluate this? Let θ equal what is in parentheses

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