Inverse Trigonometric Functions

1. Inverse Trigonometric Functions Trigonometry MATH 103 S. Rook

2. Overview • Section 4.7 in the textbook: • Review of inverse functions • Inverse sine function • Inverse cosine function • Inverse tangent function • Inverse trigonometric functions and right triangles

3. Review of Inverse Functions

4. Review of Inverse Functions • Graphically, a function f has an inverse if it passes the horizontal line test f is said to be one-to-one • Given a function f, let f-1 be the relation that results when we swap the x and y coordinates for each point in f • If f and f-1 are inverses, their domains and ranges are interchanged: • i.e. the domain of f becomes the range of f-1 & the range of f becomes the domain of f-1 and vice versa

5. Review of Inverse Functions (Continued) • None of the six trigonometric functions have inverses as they are currently defined • All fail the horizontal line test • We will examine how to solve this problem soon

6. Inverse Sine Function

7. Inverse Sine Function • As mentioned earlier, y = sin x has no inverse because it fails the horizontal line test • However, if we RESTRICT the domain of y = sin x, we can force it to be one-to-one • A common domain restriction is • The restricted domain now passes the horizontal line test

8. Inverse Sine Function (Continued) • The inverse function of y = sin x is y = sin-1x • Switch all (x, y) pairs in the restricted domain of y = sin x • A COMMON MISTAKE is to confuse the inverse notation with the reciprocal • To avoid confusion, y = sin-1x is often written as y = arcsin x • Pronounced “arc sine” • Be familiar with BOTH notations

9. Inverse Sine Function (Continued) • For the restricted domain of y = sin x: D: [-π⁄2, π⁄2]; R: [-1, 1] • Then for y = arcsin x: D: [-1, 1]; R: [- π⁄2, π⁄2] • Recall that functions and their inverses swap domain and range • This corresponds to angle in either QI or QIV y = sin-1x and y = arcsin x both mean x = sin y • i.e. y is the angle in the interval [- π⁄2, π⁄2] whose sine is x

10. Inverse Sine Function (Example) Ex 1: Evaluate if possible without using a calculator – leave the answer in radians: a) b) arcsin(-2)

11. Inverse Sine Function (Example) Ex 2: Evaluate if possible using a calculator – leave the answer in degrees:

12. Inverse Cosine Function

13. Inverse Cosine Function • As mentioned earlier, y = cos x has no inverse because it fails the horizontal line test • However, if we RESTRICT the domain of y = cos x, we can force it to be one-to-one • A common domain restriction is • The restricted domain now passes the horizontal line test

14. Inverse Cosine Function (Continued) • The inverse function of y = cos x is y = cos-1x • Switch all (x, y) pairs in the restricted domain of y = cos x • To avoid confusion, y = cos-1x is often written as y = arccos x • Pronounced “arc cosine” • Be familiar with BOTH notations

15. Inverse Cosine Function (Continued) • For the restricted domain of y = cos x: D: [0, π]; R: [-1, 1] • Then for y = arccos x: D: [-1, 1]; R: [0, π] • This corresponds to an angle in either QI or QII y = cos-1x and y = arccos x both mean x = cos y • i.e. y is the angle in the interval [0, π] whose cosine is x

16. Inverse Cosine Function (Example) Ex 3: Evaluate if possible without using a calculator – leave the answer in radians: a) arccos(-3⁄2) b) cos-1(1)

17. Inverse Tangent Function

18. Inverse Tangent Function • As mentioned earlier, y = tan x has no inverse because it fails the horizontal line test • However, if we RESTRICT the domain of y = tan x, we can force it to be one-to-one • A common domain restriction is • The restricted domain now passes the horizontal line test

19. Inverse Tangent Function (Continued) • The inverse function of y = tan x is y = tan-1x • Switch all (x, y) pairs in the restricted domain of y = tan x • To avoid confusion, y = tan-1x is often written as y = arctan x • Pronounced “arc tangent” • Be familiar with BOTH notations

20. Inverse Tangent Function (Continued) • For the restricted domain of y = tan x: D: [-π⁄2, π⁄2]; R: (-oo, +oo) • Then for y = arctan x: D: (-oo, +oo); R: [-π⁄2, π⁄2] • This corresponds to an angle in either QI or QIV y = tan-1x and y = arctan x both mean x = tan y • i.e. y is the angle in the interval [-π⁄2, π⁄2] whose tangent is x

21. Inverse Tangent Function (Example) Ex 4: Evaluate if possible without using a calculator – leave the answer in radians:

22. Inverse Trigonometric Functions and Right Triangles

23. Taking the Inverse of a Function • Recall what happens when we take the inverse of a function: • e.g. Given x = 3, because y = ln x and ex are inverses: • In other words, we get the original argument PROVIDED that the argument lies in the domain of the function AND its inverse • This also applies to the trigonometric functions and their inverse trigonometric functions

24. Inverse Trigonometric Functions and Right Triangles • The same technique does not work when the functions are NOT inverses • E.g. tan(sin-1x) • Recall the meaning of sin-1x • i.e. the sine of what angle results in x • With this information, we can construct a right triangle using Definition II of the Trigonometric functions • We can use the right triangle to find

25. Inverse Trigonometric Functions and Right Triangles (Example) Ex 5: Evaluate without using a calculator: a) b) c) d)

26. Inverse Trigonometric Functions and Right Triangles (Example) Ex 6: Write an equivalent expression that involves x only – assume x is positive:

27. Summary • After studying these slides, you should be able to: • State whether or not an argument falls in the domain of the inverse sine, inverse cosine, or inverse tangent • Evaluate the inverse trigonometric functions both by hand or by calculator • Evaluate expressions using inverse trigonometric functions • Additional Practice • See the list of suggested problems for 4.7 • Next lesson • Proving Identities (Section 5.1)