Inverse Trig Functions
6.1
JMerrill, 2007
Revised 2009
Recall
• From College Algebra, we know that
for a function to have an inverse that
is a function, it must be one-to-one—
it must pass the Horizontal Line
Test.
Sine Wave
• From looking at a sine wave, it is
obvious that it does not pass the
Horizontal Line Test.
Sine Wave
• In order to pass the Horizontal Line
Test (so that sin x has an inverse
that is a function), we must restrict
the domain.
• We restrict it
to   ,  
 2 2 
Sine Wave
  
 2 , 0 
• Quadrant IV is
 
• Quadrant I is 0, 2 
• Answers must be in one of those two
quadrants or
the answer
doesn’t exist.
Sine Wave
• How do we draw inverse functions?
• Switch the x’s and y’s!
Switching the x’s and y’s also
means switching the axis!
Sine Wave
• Domain/range of restricted wave?
• Domain/range of inverse? D :  1,1
   
R:
, 
 2 2
   
D:
, 
 2 2
R :  1,1
Inverse Notation
• y = arcsin x
or
y = sin-1 x
• Both mean the same thing. They
mean that you’re looking for the
angle (y) where sin y = x.
Evaluating Inverse
Functions
• Find the exact value of:
• Arcsin ½
– This means at what angle is the sin = ½ ?
– π/6
– 5π/6 has the same answer, but falls in
QIII, so it is not correct.
Calculator
• When looking for an inverse answer
on the calculator, use the 2nd key
first, then hit sin, cos, or tan.
• When looking for an angle always hit
the 2nd key first.
• Last example: Degree mode, 2nd, sin,
.5 = 30.
Evaluating Inverse
Functions
• Find the value of:
• sin-1 2
– This means at what angle is the sin = 2 ?
– What does your calculator read? Why?
– 2 falls outside the range of a sine wave
and outside the domain of the inverse
sine wave
Cosine Wave
Cosine Wave
• We must restrict the domain
• Now the inverse
D :  1,1
R :  0,  
D :  0,  
R :  1,1
Cosine Wave
 
0, 2 
• Quadrant I is
 
• Quadrant II is  2 ,  
• Answers must be in one of those two
quadrants or
the answer
doesn’t exist.
Tangent Wave
Tangent Wave
• We must restrict the domain
• Now the inverse
Graphing Utility: Graph the following inverse functions.
Set calculator to radian mode.
a. y = arcsin x

–1.5
1.5
–
2
b. y = arccos x
–1.5
1.5
–

c. y = arctan x
–3
3
–
Graphing Utility: Approximate the value of each expression.
Set calculator to radian mode.
a. cos–1 0.75
b. arcsin 0.19
c. arctan 1.32
d. arcsin 2.5
Composition of Functions
• Find the exact value of
• sin  sin 1 2 

2 
2
2

4
• Where is the sine =
• Replace the parenthesis in the
original problem with that answer
2
• Now solve sin 
4
2
Example
• Find the exact value of
3 

sin 1  sin 
4 

• The sine angles must be in QI or
QIV, so we must use the reference

angle
4
2
1
•
3 

sin
1 
1 
sin  sin   sin  sin 
4 
4



2
sin 
4
2

4
2
Example
• Find tan(arctan(-5))
-5
• Find

1  1 
cos  cos

2

1
2
• If the words are the same and the
inverse function is inside the
parenthesis, the answer is already
given!
Example
2

tan  arccos 
3

• Find the exact value of
• Steps:
• Draw a triangle using only the info
inside the parentheses.
3
5
• Now use your x, y, r’s
2
to answer the outside term
y
5
ta n   
x
2
Last Example

1 7 
tan  cos

12


• Find the exact value of
• Cos is negative in QII and III, but
the inverse is restricted to QII.
y  95
tan   
x
7
95
12
-7
You Do
• Find the exact value of
 1 3 
tan  sin

7 

3 10
20