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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 oneto-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 3 8 b. Cos -1 (-.2) Example: Find without using a calc: a. Tan -1 (–1) b. Tan -1 (1) c. Sin -1 2 2 d. Cos 2 -1 2 Examples: - Find sec (Tan -1 1 3 ) with a calc. 4 - Find csc (Cos ( )) using a graph 5 (without a calc.). -1 Example: - Find the approximate value and exact value of csc (Cos -1 ( - .4)).