The Inverse Sine, Cosine, and Tangent Functions Section 7.1 Mr. Fiscina “The most certain way to succeed, is to try just one more time.” ~Thomas Edison

Inverses What do you know about inverses? What does one-to-one mean? What is the horizontal line test?

The inverse Sine Function JUST KIDDING

But seriously… Graph the sine function [𝑦=sin⁡(𝑥)] Which horizontal lines pass through the graph? Do they pass through the graph more than once? What does that mean?

The inverse sine function However, if we restrict the domain of 𝑦= sin 𝑥 to the interval [− 𝜋 2 , 𝜋 2 ], the restricted function is one-to-one and so will have an inverse function.

Find the exact value of an inverse sine function For some numbers x, it is possible to find the exact value of 𝑦= 𝑠𝑖𝑛 −1 𝑥. Find the exact value of: sin −1 1 Reminder, we are restricted to [− 𝜋 2 , 𝜋 2 ]. More examples, find sin -1 (-1/2) Approximate sin -1 (1/3) Aprroximate sin -1 (-1/4)

Use properties of inverse functions to find exact values of certain composite functions 𝑓 −1 𝑓 𝑥 = sin −1 sin 𝑥 =𝑥 𝑤ℎ𝑒𝑟𝑒 − 𝜋 2 ≤𝑥≤ 𝜋 2 𝑓 𝑓 −1 𝑥 = sin sin −1 𝑥 =𝑥 𝑤ℎ𝑒𝑟𝑒 −1≤𝑥≤1 Find the exact value of each of the following composite functions: sin −1 ( sin 𝜋 8 ) sin −1 ( sin 5𝜋 8 ) 5 pi over 8 is not in the interval with the pi over twos…therefore we need to find something in that interval.

Practice for inverse sine functions Page 451, numbers 13, 15, 19, 22, 24, 25, 30, 38, 40, 41.

The inverse cosine function Cosine has a different restriction than sine. What is it? Why is it that?

The inverse cosine function

Find the exact value of an inverse cosine function 𝑦= cos −1 𝑥 𝑚𝑒𝑎𝑛𝑠 𝑥= cos 𝑦 Where 1≤𝑥≤−1 and 0≤𝑦≤𝜋 Example: Find the exact value of: cos −1 0 Example: Find the exact value of: cos −1 (− 2 2 )

Composition of cosine functions cos −1 ( cos 𝑥) =𝑥 Where 0≤𝑥≤𝜋 cos cos −1 𝑥 =𝑥 Where −1≤𝑥≤1 Example: Find the exact value of cos −1 ( cos 𝜋 12 ) Example: Find the exact value of cos [ cos −1 (−0.4)] Example: “ “ cos −1 [ cos (− 2𝜋 3 )] Example: “ “ cos ( cos −1 𝜋 )

The inverse tangent function What restriction should this function have, if any? Why does your response make sense? What are you having for dinner later?

Turn to page 448, We’re going to do example 9 together…I don’t feel like typing anymore.

Ok Close your books, I can type again Find the inverse function of 𝑓 𝑥 =2 sin 𝑥 −1. What are the domain and range for both the function and its inverse?

Solving 3 sin −1 𝑥 =𝜋

Page 451 13-67 Odds (for numbers 13-44 skip the problems about sine) Want to do a couple together?