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Inverse Trig Functions 6.1

Inverse Trig Functions 6.1. JMerrill, 2007 Revised 2009. Recall. From College Algebra, we know that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test. Sine Wave.

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Inverse Trig Functions 6.1

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  1. Inverse Trig Functions6.1 JMerrill, 2007 Revised 2009

  2. Recall • From College Algebra, we know that for a function to have an inverse that is a function, it must be one-to-one—it must pass the Horizontal Line Test.

  3. Sine Wave • From looking at a sine wave, it is obvious that it does not pass the Horizontal Line Test.

  4. Sine Wave • In order to pass the Horizontal Line Test (so that sin x has an inverse that is a function), we must restrict the domain. • We restrict it to

  5. Sine Wave • Quadrant IV is • Quadrant I is • Answers must be in one of those two quadrants or the answer doesn’t exist.

  6. Sine Wave • How do we draw inverse functions? • Switch the x’s and y’s! Switching the x’s and y’s also means switching the axis!

  7. Sine Wave • Domain/range of restricted wave? • Domain/range of inverse?

  8. Inverse Notation • y = arcsin x or y = sin-1 x • Both mean the same thing. They mean that you’re looking for the angle (y)where sin y = x.

  9. Evaluating Inverse Functions • Find the exact value of: • Arcsin ½ • This means at what angle is the sin = ½ ? • π/6 • 5π/6 has the same answer, but falls in QIII, so it is not correct.

  10. Calculator • When looking for an inverse answer on the calculator, use the 2nd key first, then hit sin, cos, or tan. • When looking for an angle always hit the 2nd key first. • Last example: Degree mode, 2nd, sin, .5 = 30.

  11. Evaluating Inverse Functions • Find the value of: • sin-1 2 • This means at what angle is the sin = 2 ? • What does your calculator read? Why? • 2 falls outside the range of a sine wave and outside the domain of the inverse sine wave

  12. Cosine Wave

  13. Cosine Wave • We must restrict the domain • Now the inverse

  14. Cosine Wave • Quadrant I is • Quadrant II is • Answers must be in one of those two quadrants or the answer doesn’t exist.

  15. Tangent Wave

  16. Tangent Wave • We must restrict the domain • Now the inverse

  17. Graphing Utility: Graphs of Inverse Functions  –1.5 1.5 – 2 –1.5 1.5 –  –3 3 – Graphing Utility:Graph the following inverse functions. Set calculator to radian mode. a. y = arcsin x b. y = arccos x c. y = arctan x

  18. Graphing Utility: Inverse Functions Graphing Utility:Approximate the value of each expression. Set calculator to radian mode. a. cos–1 0.75 b. arcsin 0.19 c. arctan 1.32 d. arcsin 2.5

  19. Composition of Functions • Find the exact value of • Where is the sine = • Replace the parenthesis in the original problem with that answer • Now solve

  20. Example • Find the exact value of • The sine angles must be in QI or QIV, so we must use the reference angle

  21. Example • Find tan(arctan(-5)) -5 • Find • If the words are the same and the inverse function is inside the parenthesis, the answer is already given!

  22. Example • Find the exact value of • Steps: • Draw a triangle using only the info inside the parentheses. • Now use your x, y, r’s to answer the outside term 3 2

  23. Last Example • Find the exact value of • Cos is negative in QII and III, but the inverse is restricted to QII. 12 -7

  24. You Do • Find the exact value of

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