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?
We find inverses by interchanging the x and y values.
• Are inverses of functions always functions?
No
• How did we test for this?
Vertical line test.
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?
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
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
y = sin x
Range
y = Sin x
y = sin x
y = Sin x
y = cos x
all real numbers
y = Cos x
y = cos x
y = Cos x
y = tan x
all real numbers
except n,
y = Tan x
y = tan x
y = Tan x
all real numbers
all real numbers
all real numbers
where n is an odd
integer
Complete the following table on your own
Function
Domain
Range
y = Sin x
y = Arcsin x
1  x  1


2
 y

2
y = Cos x
y = Arccos x
1  x  1
y = Tan x
y = Arctan x
0 y 
all real numbers
all real numbers


2
 y

2
Table of Values of Sin x and Arcsin x
y = Sin x
X
y = Arcsin x
Y
X
Y
-π/2
-π/2
-π/6
-π/6
0
0
π/6
π/6
π/2
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
X
y = Arccos x
Y
X
Y
0
0
π/3
π/3
π/2
π/2
2π/3
2π/3
π
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 = Arctan x
Y
X
Y
-π/2
-π/2
-π/4
-π/4
0
0
π/4
π/4
π/2
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 y
2. Solve for y
Let’s graph 2Tan x = y first.
Complete the table:
y = 2Tan x
X
x = Arctan ½y
Tan x = ½y
2Tan x = y
Then graph!
-π/2
-π/4
0
π/4
Now graph the original
function, y = Arctan ½x by
switching the table you just
completed!
π/2
Y
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
Let’s graph y = Sin(2x) first.
Why are these x-values used?
y = Sin2x
X
-π/4
-π/12
0
Now graph the inverse
function, y = Arcsin(x)/2 by
switching the table you just
completed!
π/12
π/4
Y
Evaluate each expression
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.