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 numbersRange: [-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 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

Domain Range

y = sin x

all real numbers

y = Sin x y = sin x y = Sin x

y = cos xall real

numbers

y = Cos x y = cos x y = Cos x

y = tan xall real

numbers except n,

where n is an odd integer

y = Tan x y = tan x

all real numbers

y = Tan x

all real numbers

Complete the following table on your own

Function Domain Range

y = Sin x

y = Arcsin x

y = Cos x

y = Arccos x

y = Tan xall real numbers

y = Arctan x

Table of Values of Sin x and Arcsin x

y = Sin x

X Y

-π/2

-π/6

0

π/6

π/2

y = Arcsin x

X Y

-π/2

-π/6

0

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

0

π/3

π/2

2π/3

π

y = Arccos x

X Y

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

X Y

-π/2

-π/4

0

π/4

π/2

y = Arctan x

X Y

-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.

To write the equation:1.Exchange x and y2.Solve for y

x = Arctan ½yTan x = ½y2Tan x = y

Let’s graph 2Tan x = y first.Complete the table:

Then graph!

y = Tan x

X Y

-π/2

-π/4

0

π/4

π/2Now 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.

To write the equation:1.Exchange x and y2.Solve for y

x = Sin(2y)Arcsin(x) = 2yArcsin(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!

y = Sin2x

X Y

-π/4

-π/12

0

π/12

π/4

Evaluate each expression

Evaluate each expression