7.6 – The Inverse Trigonometric Ratios Essential Question: How do you make a function without an inverse have an inverse?

From chapter 4, we learned that for a function to be one-to-one and have an inverse it must pass the horizontal line test.

If you look at the tangent graph, the graph of tan x is not one-to-one, so therefore has no inverse. However, if we restrict our domain, like in the second picture, it is one- to-one and thus has an inverse.

Tangent Graph The inverse is the third picture and is denoted: f -1 (x) = Tan -1 x

Example Find Tan -1 3 with a calc in a) degrees and b) radians.

The inverse of the sine and cosine graphs: Pictures:

Example: Find with calc: a. Sin -1 b. Cos -1 (-.2)

Example: Find without using a calc: a. Tan -1 (–1) b. Tan -1 (1) c. Sin -1 d. Cos -1

Examples: - Find sec (Tan -1 ) with a calc. - Find csc (Cos -1 ( )) using a graph (without a calc.).

Example: - Find the approximate value and exact value of csc (Cos -1 ( -.4)).