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 y = sin x y = cos x y = tan x y = sin -1 x y = cos -1 x y = tan -1 x
Inverse y = arcsin x y = arccos x y = arctan x

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

B. Now let’s look at y = cos x

What is the domain and range of cos x?
Domain: all real numbers
Range: [-1, 1]

What would the graph of y = arccos x look like?

What is the domain and range of arccos x?
Domain: [-1, 1]
Range: all real numbers

C. Now let’s look at y = tan x

What is the domain and range of tan x?

What would the graph of y = arctan x look like?

What is the domain and range of arctan x?

D. Are the inverses of sin x, cos x, and tan x functions?


However, we can make them functions by restricting their domains.
Capital letters are used to distinguish when the function’s domain is restricted.
Inverse Function Original Functions with
Restricted Domain y = Sin x y = Cos x y = Tan x y = Sin -1 x y = Cos -1 x y = Tan -1 x y = Arcsin x y = Arccos x y = Arctan x

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

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

II. Graphing with Restricted Domains
A. Table of Values of Sin x and Arcsin x
X
π/2
-π/6
0
π/6
π/2 y = Sin x
Y
X y = Arcsin x
Y
π/2
-π/6
0
π/6
π/2
Why are we using these values?

II. Graphing with Restricted Domains
A. Table of Values of Sin x and Arcsin x
X
π/2
-π/6
0
π/6
π/2 y = Sin x
Y
-1
-0.5
0
0.5
1
X
-1
-0.5
0
0.5
1 y = Arcsin x
Y
π/2
-π/6
0
π/6
π/2
Why are we using these values?

Graphs of Sin x and Arcsin x

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

Table of Values of Cos x and Arccos x
X
0
π/3
π/2
2π/3
π y = Cos x
Y
1
0.5
0
-0.5
-1
Why are we using these values?
X
1
0.5
0
-0.5
-1 y = Arccos x
Y
0
π/3
π/2
2π/3
π

Graphs of Cos x and Arccos x

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

Table of Values of Tan x and Arctan x
-
X
π/2
-π/4
0
π/4
π/2 y = Tan x
Y
(veritical asymptote)
-1
0
1
(vertical asymptote)
Why are we using these values?
y = Arctan x
X
(horz asymptote)
-1
0
1
(horizontal asymptote)
Y
π/2
-π/4
0
π/4
π/2

Graphs of Tan x and Arctan x

III. Writing and graphing Inverse Trig Functions
Ex 1. 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 y
2. Solve for y x = Arctan ½y
Tan x = ½y
2Tan x = y

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

Ex 2: 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 y
2. Solve for y x = Sin(2y )
Arcsin(x) = 2y
Arcsin(x)/2 = y

Write an equation for the inverse of y = Sin(2x).
Then graph the function and its inverse.
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!
X y = Sin2x
Y
-π/4
-π/12
0
π/12
π/4
X y = Sin2x
Y

IV. Evaluate each expression
See hand-written notes