4.7 Inverse Trigonometric Functions

1. 4.7 Inverse Trigonometric Functions

2. Inverse functions • g(x) is the inverse function of f(x) IF g(f(x)) = x and f(g(x)) = x We notate an inverse function as f-1(x) Example f(x) = 4x f-1(x)=

3. Remember your favorite inverse functions? Logarithms and exponentials? f(x) = 2x f-1(x)= log2x

4. Starboard demo NOOO!! It does not have an inverse function Does it pass the horizontal line test? Does it pass the vertical line test? Yes, so it’s a function!! Restrict the domain to f(x)= x2 , x ≥ 0 Now it passes the horizontal line test.

5. Consider the graph f(x) = sinx Is it one-to-one?

6. y x y = sin x Inverse Sine Function Inverse Sine Function Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f(x) = sin x does not pass the Horizontal Line Test IT MUST BE RESTRICTED!! Sin x has an inverse function on this interval.

7. This is another way to write arcsin x. Inverse Sine Function The inverse sine function is defined by y = arcsin x if and only if sin y = x. The domain of y = arcsin x is [–1, 1]. The range of y = arcsin x is [–/2 , /2]. Example:

8. Cartoon time • Oh, sine machine. He is soo happy outputting side to side ratios…. Takes in angles- outputs side to side ratios… BUT, when his arch enemy ARCSINE comes along, he has to fight the guy who undoes everything he does.

9. y x y = cos x Inverse Cosine Function Inverse Cosine Function f(x) = cos x must be restricted to find its inverse. Cos x has an inverse function on this interval.

10. Angle whose cosine is x This is another way to write arccos x. Inverse Cosine Function The inverse cosine function is defined by y = arccos x if and only if cos y = x. The domain of y = arccos x is [–1, 1]. The range of y = arccos x is [0 , ]. Example:

11. y = tan x y x Inverse Tangent Function Inverse Tangent Function f(x) = tan x must be restricted to find its inverse. Tan x has an inverse function on this interval.

12. Angle whose tangent is x The domain of y = arctan x is . This is another way to write arctan x. Inverse Tangent Function The inverse tangent function is defined by y = arctan x if and only if tan y = x. The range of y = arctan x is [–/2 , /2]. Example:

13. Examples

14. Consider a slightly different setup: This is also the composition of two inverse functions but… Did you suspect the answer was going to be 120 degrees? This problem behaved differently because the first angle, 120 degrees, was outside the range of the arcsin. So use some caution when evaluating the composition of inverse trig functions. The remainder of this presentation consists of practice problems, their answers and a few complete solutions.

15. Warm-up Find the six trig functions of Ө=30o Triangle A Ө Triangle B 30

16. Page 328 #1-4; 6-8; 13-20; 37; 40; 49-55; 71

17. Restricted domain • How to tell if a function has an inverse function

18. Examples

19. For a function to have an inverse function, it has to be one-to-one Does it pass the horizontal line test? NOOO!! It does not have an inverse function Does it pass the vertical line test? Yes, so it’s a function!!