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
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.
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
E. 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
F. 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
II. Graphing with Restricted Domains
A. 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?
II. Graphing with Restricted Domains
A. Table of Values of Sin x and Arcsin x
y = Sin x
y = Arcsin x
X
Y
X
Y
-π/2
-1
-1
-π/2
-π/6
-0.5
-0.5
-π/6
0
0
0
0
π/6
0.5
0.5
π/6
π/2
1
1
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
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?
Table of Values of Cos x and Arccos x
y = Cos x
y = Arccos x
X
Y
X
Y
0
1
1
0
π/3
0.5
0.5
π/3
π/2
0
0
π/2
2π/3
-0.5
-0.5
2π/3
π
-1
-1
π
Why are we using these values?
Graphs of Cos x and Arccos x
C. 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?
Table of Values of Tan x and Arctan x
y = Tan x
y = Arctan x
X
Y
X
Y
-π/2
(veritical
asymptote)
(horz asymptote)
-π/2
-π/4
-1
-1
-π/4
0
0
0
0
π/4
1
1
π/4
π/2
(vertical
asymptote)
(horizontal
asymptote)
π/2
Why are we using these values?
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:
y = 2Tan x
X
Y
-π/2
Then graph!
-π/4
0
π/4
π/2
y = Arctan ½ x
Now graph the
original function,
y = Arctan ½x by
switching the table
you just completed!
X
Y
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?
y = Sin2x
X
Y
-π/4
-π/12
0
π/12
π/4
y = Sin2x
Now graph the inverse
function, y = Arcsin(x)/2 by
switching the table you just
completed!
X
Y
IV. Evaluate each expression
See hand-written notes