Inverse Trigonometric Functions

1. Inverse Trigonometric Functions • Recall some facts about inverse functions: • For a function to have an inverse it must be a one-to-one function. • The domain of f -1is the range off. • The range of f -1is the domain off . • If (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f -1. • f(f -1) = x for all x in the domain of f -1. • f -1 (f ) = x for all x in the domain of f .

2. Note that it fails the horizontal line test. • The graph of y = sin x is given below. • Since y = sin x is not a 1-1 function on its domain of all real numbers, it does not have an inverse.

3. We must restrict the domain of the sine function so that it will be a 1-1 function, and have an inverse. • This could be done in several ways, but the following is most common:

4. Definition: Inverse Sine

5. Definition: Inverse Sine • Another way of writing the inverse of the sine function is as follows:

6. Note that it fails the horizontal line test. • The graph of y = cosx is given below. • Since y = cosx is not a 1-1 function on its domain of all real numbers, it does not have an inverse.

7. We must restrict the domain of the cosine function so that it will be a 1-1 function, and have an inverse. • This could be done in several ways, but the following is most common:

8. Definition: Inverse Cosine

9. Definition: Inverse Cosine • Another way of writing the inverse of the cosine function is as follows:

10. Note that it fails the horizontal line test. • The graph of y = tan x is given below. • Since y = tanx is not a 1-1 function on its normal domain, it does not have an inverse.

11. We must restrict the domain of the tangent function so that it will be a 1-1 function, and have an inverse. • This could be done in several ways, but the following is most common:

12. Definition: Inverse Tangent

13. Definition: Inverse Tangent • Another way of writing the inverse of the tangent function is as follows: