Warm up Find the values of θ for which cot θ = 1 is true. Write the equation for a tangent function whose period is 4π, phase shift 0, and vertical shift +3.

LESSON 6-8 – TRIG INVERSES AND THEIR GRAPHS Objective: To graph inverse trigonometric functions. To find principal values of inverse trig functions.

Quick Review How do you find inverses of functions? Are inverses of functions always functions? How did we test for this? We find inverses by interchanging the x and y values. No Vertical line test.

Inverse Trig Functions Original Function Inverse y = sin xy = sin -1 xy = arcsin x y = cos xy = cos -1 xy = arccos x y = tan xy = tan -1 xy = arctan x

Consider the graph of y = sin x  What is the domain and range of sin x?  What would the graph of y = arcsin x look like?  What is the domain and range of arcsin x? Domain: all real numbers Range: [-1, 1] Domain: [-1, 1] Range: all real numbers

Is the inverse of sin x a function? This will also be true for cosine and tangent. Therefore all of the domains are restricted in order for the inverses to be functions.

How do you know if the domain is restricted for the original functions? Capital letters are used to distinguish when the function’s domain is restricted. Original Functions with Restricted Domain Inverse Function y = Sin xy = Sin -1 xy = Arcsin x y = Cos xy = Cos -1 xy = Arccos x y = Tan xy = Tan -1 xy = Arctan x

Original Domains  Restricted Domains DomainRange y = sin x all real numbers y = Sin xy = sin xy = Sin x y = cos x all real numbers y = Cos xy = cos xy = Cos x y = tan x all real numbers except n, where n is an odd integer y = Tan xy = tan x all real numbers y = Tan x all real numbers

Complete the following table on your own FunctionDomainRange y = Sin x y = Arcsin x y = Cos x y = Arccos x y = Tan x all real numbers y = Arctan x all real numbers

Table of Values of Sin x and Arcsin x y = Sin x XY - π/2 -π/6 0 π/6 π/2 y = Arcsin x XY - π/2 -π/6 0 π/6 π/2 Why are we using these values? Principal Values – values of Sine, Cosine etc. when the domain is restricted.

Graphs of Sin x and Arcsin x

Table of Values of Cos x and Arccos x y = Cos x XY 0 π/3 π/2 2π/3 π y = Arccos x XY 0 π/3 π/2 2π/3 π Why are we using these values?

Graphs of Cos x and Arccos x

Table of Values of Tan x and Arctan x y = Tan x XY - π/2 -π/4 0 π/4 π/2 y = Arctan x XY - π/2 -π/4 0 π/4 π/2 Why are we using these values?

Graphs of Tan x and Arctan x

Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse. y = 2Tan x XY - π/2 -π/4 0 π/4 π/2 To write the equation: 1.Exchange x and y 2.Solve for y x = Arctan ½y Tan x = ½y 2Tan x = y Let’s graph 2Tan x = y first. Complete the table: Then graph! Now graph the original function, y = Arctan ½x by switching the table you just completed!

Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse. y = Sin2x XY -π/4 -π/12 0 π/12 π/4 To write the equation: 1.Exchange x and y 2.Solve for y x = Sin(2y ) Arcsin(x) = 2y Arcsin(x)/2 = y Let’s graph y = Sin(2x) first. Why are these x-values used? Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!

Evaluate each expression

Example Determine whether Sin -1 (sin x)=x is true or false for all values of x. If false, give a counterexample.