4.7 Inverse Trig Functions

1. 4.7Inverse Trig Functions JMerrill, 2010

2. Recall • We know that for a function to have an inverse 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. The Unit Circle • The Unit Circle in found in section 4.7. • We will use: • Radians • Exact answers (mostly) • Quick board review of Unit Circle, quadrants on the wave, & converting to radian measure

5. 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

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

7. 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!

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

9. Inverse Notation • y = arcsin x or y = sin-1 x • Both mean the same thing. They mean that you’re looking for the anglewhere sin y = x.

10. 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)

11. 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. • Radian mode: 2nd, sin, .5 = .524 (which is pi/6)

12. 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 domain of an inverse sine wave

13. Cosine Wave

14. Cosine Wave • Domain and range of restricted wave? • Domain and range of the inverse?

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

16. Tangent Wave

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

18. 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

19. 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

20. Composition of Functions • Previously learned notation: • fog(x) gof(x)

21. Composition of Functions Using Inverse Properties • Find tan(arctan(-5)) • -5 (the tangent and its inverse cancel each other out!) • Find arcsin(sin ) • The domain of the sine function is . Since is outside that domain, we’ll just say that the answer is: is outside the domain (unless you remember coterminal angles and can tell me the actual answer is • Find • Outside the domain

22. Composition of Functions • Find the exact value of • Draw the graph using only the info inside the parentheses. (Easy way—completely ignore the fact that you have inverses!)

23. y x Example: Evaluating Composition of Functions Example: Positive so draw in Q1) 3 2

24. You Try • Find the