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? 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
y = Cos x
y = Arccos x
y = Tan x
y = Arctan x
all real numbers
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?
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 = Tan 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