6.8 – Trig Inverses and their graphs

Quick Review How do you find inverses of functions? Are inverses of functions always functions? How did we test for this?

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

Original Functions with Restricted Domain 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 x y = Sin-1 x y = Arcsin x y = Cos x y = Cos-1 x y = Arccos x y = Tan x y = Tan-1 x y = Arctan x

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

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

Table of Values of Sin x and Arcsin x y = Sin x X Y -π/2 -π/6 π/6 π/2 y = Arcsin x X Y -π/2 -π/6 π/6 π/2 Why are we using these values?

Graphs of Sin x and Arcsin x

Table of Values of Cos x and Arccos x y = Cos x X Y π/3 π/2 2π/3 π y = Arccos x X Y 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 X Y -π/2 -π/4 π/4 π/2 y = Arctan x X Y -π/2 -π/4 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 Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse. To write the equation: Exchange x and y Solve for y Let’s graph 2Tan x = y first. Complete the table: Then graph! y = Tan x X Y -π/2 -π/4 π/4 π/2 x = Arctan ½y Tan x = ½y 2Tan x = y 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) Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse. To write the equation: Exchange x and y Solve for y Let’s graph y = Sin(2x) first. Why are these x-values used? y = Sin2x X Y -π/4 -π/12 π/12 π/4 x = Sin(2y) Arcsin(x) = 2y Arcsin(x)/2 = y Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!

Evaluate each expression

Evaluate each expression