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Chapter 4
Trigonometric
Functions
4.7 Inverse Trigonometric
Functions
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Objectives:
Understand and use the inverse sine function.
Understand and use the inverse cosine function.
Understand and use the inverse tangent function.
Use a calculator to evaluate inverse trigonometric
functions.
Find exact values of composite functions with inverse
trigonometric functions.
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Inverse Functions
Here are some helpful things to remember from our
earlier discussion of inverse functions:
If no horizontal line intersects the graph of a function
more than once, the function is one-to-one and has an
inverse function.
If the point (a, b) is on the graph of f, then the point
(b, a) is on the graph of the inverse function, denoted
f –1. The graph of f –1 is a reflection of the graph of f
about the line y = x.
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The Inverse Sine Function
The horizontal line test shows that the sine function
is not one-to-one; y = sin x has an inverse function on
the restricted domain    x   .
2
2
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The Inverse Sine Function (continued)
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Graphing the Inverse Sine Function
One way to graph y = sin–1 x is to take points on the graph
of the restricted sine function and reverse the order of the
coordinates.
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Graphing the Inverse Sine Function (continued)
Another way to obtain the graph of y = sin–1 x is to reflect
the graph of the restricted sine function about the line y = x.
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Finding Exact Values of sin–1x
1. Let   sin 1 x.


  .
2
2
3. Use the exact values in the table to find the value of 
in    ,   that satisfies sin   x.
 2 2 
2. Rewrite   sin x as sin  x, where 
1
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Example: Finding the Exact Value of an Inverse Sine
Function
Find the exact value of sin
1
3
.
2
Step 1 Let   sin 1 x .
3
  sin
2


1
Step 2 Rewrite   sin x as sin  x, where     .
2
2
3
sin  
2
1
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Example: Finding the Exact Value of an Inverse Sine
Function (continued)
Find the exact value of sin
1
3
.
2
Step 3 Use the exact value in the table to find the value
of  in    ,   that satisfies sin  x .
 2 2 

The angle in    ,   whose sine is 3 is .
 2 2 
3
2
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Example: Finding the Exact Value of an Inverse Sine
Function

2
Find the exact value of sin    .
 2 
Step 1 Let   sin 1 x .
2
1 
  sin   
 2 


1
Step 2 Rewrite   sin x as sin  x, where     .
2
2
2
sin   
2
1
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Example: Finding the Exact Value of an Inverse Sine
Function (continued)

2
Find the exact value of sin    .
 2 
Step 3 Use the exact value in the table to find the value
of  in    ,   that satisfies sin  x .
 2 2 
1

The angle in    ,   whose sine is  2 is  .
 2 2 
4
2
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The Inverse Cosine Function
The horizontal line test shows that the cosine function
is not one-to-one. y = cos x has an inverse function on
the restricted domain 0, .
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The Inverse Cosine Function (continued)
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Graphing the Inverse Cosine Function
One way to graph y = cos–1 x is to take points on the
graph of the restricted cosine function and reverse the
order of the coordinates.
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Finding Exact Values of cos–1 x
1. Let   cos 1 x.
2. Rewrite   cos 1 x as cos  x, where 0     .
3. Use the exact values in the table to find the value of 
in  0,   that satisfies cos  x.
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Example: Finding the Exact Value of an Inverse Cosine
Function
1

Find the exact value of cos    .
 2
Step 1 Let   cos 1 x .
1  1 
  cos   
 2
Step 2 Rewrite   cos 1 x as cos  x, where 0     .
1
1
cos  
2
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Example: Finding the Exact Value of an Inverse Cosine
Function (continued)
1

Find the exact value of cos    .
 2
1
Step 3 Use the exact value in the table to find the value
of  in
 0,  that satisfies
The angle in  0,  
cos  x .
2
1
whose cosine is  is
.
3
2
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The Inverse Tangent Function
The horizontal line test shows that the tangent function
is not one-to-one. y = tan x has an inverse function on
 

the restricted domain   ,  .
 2 2
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The Inverse Tangent Function (continued)
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Graphing the Inverse Tangent Function
One way to graph y = tan–1 x is to take points on the graph
of the restricted tangent function and reverse the order of
the coordinates.
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Finding Exact Values of tan–1 x
1. Let   tan 1 x.
2. Rewrite   tan x as tan   x, where 


x .
2
2
3. Use the exact values in the table to find the value of 
in    ,   that satisfies tan   x.
 2 2
1
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Example: Finding the Exact Value of an Inverse Tangent
Function
Find the exact value of tan 1  1.
Step 1 Let   tan 1 x .
  tan 1  1
Step 2 Rewrite   tan x as tan  x, where 
1

2
 
tan   1
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
2
.
Example: Finding the Exact Value of an Inverse Sine
Function (continued)
Find the exact value of tan 1  1.
Step 3 Use the exact value in the table to find the value
of  in    ,   that satisfies tan  x .


 2 2





The angle in   ,  whose tangent is –1 is  .
4
 2 2
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Graphs of the Three Basic Inverse Trigonometric
Functions
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Example: Calculators and Inverse Trigonometric
Functions
Use a calculator to find the value to four decimal places of
each function:
1
a. cos
3
1
1
cos
 1.2310
3
b. tan 1 (35.85)
1
1
tan (35.85)  1.5429
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Inverse Properties
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Example: Evaluating Compositions of Functions and
Their Inverses
Find the exact value, if possible:
a. cos(cos 1 0.7)
cos(cos 0.7)  0.7
b. sin 1 (sin  )
sin 1 (sin  )  0
1
1

cos
cos
(1.2) 
c.

–1.2 is not included in the domain of the inverse cosine
function. cos cos 1 (1.2)  is not defined.
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