1 / 24

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.

keane-donovan
Download Presentation

Inverse Trig Functions 6.1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related