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Section 5.4 - Inverse Trigonometric Functions
The Inverse Sine Function
Consider the graph of the sine function f ( x)  sin( x) .
The function is a periodic function. That means that the functions repeats its values in
regular intervals, which we call the period.
Is it one to one?
If the function is not one-to-one, we run into problems when we consider the inverse of
the function. What we want to do with the sine function is to restrict the values for sine.
When we make a careful restriction, we can get something that IS one-to-one.
   
If we limit the function to the interval  ,  , the graph will look like this:
 2 2
Restricted Sine function
  
Domain:   , 
 2 2
Range:  1,1
On this limited interval, we have a one-to-one sine function.
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Being an one-to-one function implies that the inverse of it does exist.
Notation for the inverse of sine: f ( x)  sin 1 ( x) or f ( x)  arcsin( x).
Restricted Sine Function (Blue)
Inverse Sine Function (Red)
  
Domain:   , 
 2 2
Domain:  1,1
Range:  1,1
  
Range:   ,  (quadrants 1 and 4)
 2 2
  
To find the value of sin 1 ( x ) , find the number in the interval   ,  whose sine is x.
 2 2
  1
1 
 sin 1   
Example: sin   
6 2
2 6
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The Inverse Cosine Function
Let’s do the same thing with the cosine function f ( x)  cos( x) , which is not one-to-one.
Here’s the graph of f ( x)  cos( x) .
If we limit the function to the interval  0,   , however, the function IS one-to-one.
Here’s the graph of the restricted cosine function.
Restricted Cosine Function
Domain: 0,  
Range:  1,1
On this limited interval, we have a one-to-one cosine function.
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Being an one-to-one function implies that the inverse of it does exist.
Notation for the inverse of cosine: f ( x)  cos1 ( x) or f ( x)  arccos( x).
Restricted Cosine Function (Blue)
Inverse Cosine Function (Red)
Domain: 0,  
Domain:  1,1
Range:  1,1
Range:  0,  
(quadrants 1 and 2)
To find the value of cos1 ( x) , find the number in the interval 0,   whose cosine is x.
 2 
2
 
 cos 1 
Example: cos   
 
2
4 2

 4
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The Inverse Tangent Function
Here’s the graph of tangent function f ( x)  tan( x) , which is not one-to-one:
   
,  , then the function IS one-to-one.
If we restrict the function to the interval 
 2 2
Restricted Tangent Function
  
Domain:   , 
2 2

Range:

 ,  
On this limited interval, we have a one-to-one tangent function.
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Being an one-to-one function implies that the inverse of it does exist.
1
Notation for the inverse of tangent: f ( x)  tan ( x) or f ( x)  arctan( x).
Restricted Tangent Function (Blue)
Inverse Tangent Function (Red)
  
Domain:   , 
2 2
Domain:  ,  

Range:

 ,  
   
,  (quadrants 1 and 4)
Range: 
 2 2
  
To find the value of tan 1 ( x) , find the number in the interval   ,  whose tangent is x.
2 2



 
Example: tan    1  tan 1 1 
4
4
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The Inverse Cotangent Function
Here’s the graph of cotangent function f ( x )  cot( x ) , which is not one-to-one:
If we restrict the function to the interval (0,  ) , then the function IS one-to-one.
Restricted Cotangent Function
Domain: (0,  )
Range:
 ,  
On this limited interval, we have a one-to-one cotangent function.
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Being an one-to-one function implies that the inverse of it does exist.
Notation for the inverse of cotangent: f ( x)  cot 1 ( x) or f ( x )  arc cot( x).
Restricted Cotangent Function (Blue)
Inverse Cotangent Function (Red)
Domain: (0,  )
Domain:  ,  
Range:
 ,  
Range: (0,  ) (quadrants 1 and 2)
To find the value of cot 1 ( x) , find the number in the interval (0,  ) whose cotangent is x.
 
Example: cot 1 3 

 
 cot    3
6
6
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The Inverse Secant Function
Here’s the graph of secant function f ( x )  sec( x ) , which is not one-to-one:


If we restrict the function to the interval 0,    ,   , then the function IS one-to-one.
 2 2 
Restricted Secant Function


Domain: 0,    ,  
 2 2 
Range:  ,1  1, 
On this limited interval, we have a one-to-one secant function.
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Being an one-to-one function implies that the inverse of it does exist.
Notation for the inverse of secant: f ( x)  sec 1 ( x) or f ( x)  arc sec( x).
Restricted Secant Function (Blue)
Inverse Secant Function (Red)


Domain: 0,    ,  
 2 2 
Domain:  ,1  1, 
Range:  ,1  1, 


Range: 0,    ,   (quadrants 1 and 2)
 2 2 


To find the value of sec 1 ( x ) , find the number in the interval 0,    ,  
 2 2 
whose secant is x.
Example: sec 1 2  

 
 sec   2
3
 3
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The Inverse Cosecant Function
Here’s the graph of cosecant function f ( x )  csc( x ) , which is not one-to-one:


If we restrict the function to interval  ,0    0,  , then the function IS one-to-one.
 2


2
Restricted Tangent Function


Domain:  ,0    0, 
 2


2
Range:  ,1  1, 
On this limited interval, we have a one-to-one cosecant function.
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Being an one-to-one function implies that the inverse of it does exist.
Notation for the inverse of cosecant: f ( x)  csc 1 ( x) or f ( x)  arc csc( x).
Restricted Cosecant Function (Blue)
Inverse Cosecant Function (Red)


Domain:  ,0    0, 
Domain:  ,1  1, 
Range:  ,1  1, 


Range:  ,0    0,  (quadrants 1 and 4)
 2


2
 2


2


1
To find the value of csc ( x ) , find the number in the interval  ,0    0, 
 2


2
whose cosecant is x.
Example: csc 1  2   

 
 csc    2
6
 6
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Note: We always give inverse trigonometric angles in radians.
Example 1: Compute each of the following:
1
a) sin 1  
2
b) tan 1
 3
c) arccos(0) 

d) sin 1  

2
.
2 
 1
e) sin 1   
 2
 3
.
f) cos 1 

2


g) arctan(1).
h) sec 1 (2).
i) csc 1 (0)
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NOTE: Domains of inverse trig functions:
f ( x)  sin 1 ( x) ;
[-1,1]
f ( x)  cos 1 ( x) ;
[-1,1]
f ( x)  tan 1 ( x) ;
(, )
f ( x )  cot 1 ( x ) ;
(, )
f ( x)  sec 1 ( x) ;
f ( x)  csc 1 ( x) ;
( ,1]  [1, )
( ,1]  [1, )
 
For example; sin 1 (2) or cos 1 2 are not defined.
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Composition of Trigonometric Functions with their Inverses and viceversa
Recall that if f and g are inverse functions then the following hold:
f ( g ( x ))  x for each x in the domain of g .
g ( f ( x ))  x for each x in the domain of f .
When a trigonometric function and its inverse are composed, we need to be careful about
giving an answer that is in the range of the inverse trig function.
cos1cosx   x
if x  [0,  ]
  
sin 1sin x   x if x    , 
 2 2
  
if x    , 
 2 2
tan 1tan x   x
Examples:
    
sin 1 sin   
  8  8
  7  7
but sin 1 sin   
8
  8 
    
cos 1 cos  
  8  8
  9  9
but cos 1 cos  
  8  8
    
tan 1  tan   
  8  8
  7  7
but tan 1 tan   
8
  8 
If the inverse trigonometric function is the inner function, then our job is easier.
cos[cos 1 x ]  x for any number x such that  1  x  1 .
x]  x for any number x such that
tan[tan 1 x ]  x for any number x.
sin[sin
1
1  x  1 .
Examples:

 1  1
sin sin 1   
 5  5


2
 2 
cos cos 1     
7
 7 

1 1
tan[tan 1  ] 
4 4
tan[tan 1 5]  5 .
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Example 2: Find the exact value of the following:
a)
  7
sin 1 sin 
  6

 .

  4  
b) cos 1 cos    .
  3 
c)
  3
tan 1  tan 
  4

 .

  5  
d) arcsin  sin    
  3 

 2  
e) tan 1  tan    
 3 

Example 3: Find the exact value of the following:
a)

 1 
cos cos 1    .
 6 


 5 
 
b) cos sin 1  .
13

16

 2 


c) tan cos 1     .
5


 2 
 
d) tan  cot 1    
5


 1 


e) sin  cos 1     
4


 4 


f) tan  sin 1     
5



g) tan sec 1 2 
Example 4: Simplify

 1 
cos  arctan  x  
 4 

where x  0 .
17
Working with Graphs of Inverse Trigonometric Functions
Here are the graphs of the trigonometric functions:
We can use graphing techniques learned in previous lectures to graph transformations of the
basic inverse trig functions.
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Example 5: Which of the following points is on the graph of f ( x)  arctan( x  1) ?
 
A)  ,0 
4 
 
B)  0, 
 4


C)  0, 
4

 
D)  2, 
 4
Example 6: Which of the following can be the function whose graph is given below?
A) f ( x)  cos 1 ( x  1)
B) f ( x)  sin 1 ( x  1)
C) f ( x)  cos 1 ( x  1)
D) f ( x)  sin 1 ( x  1)
E) f ( x)  tan 1 ( x  1)
Example 7: Which of the following can be the function whose graph is given below?
A) f ( x)  cos 1 ( x  2)
B) f ( x)  sin 1 ( x  2)
C) f ( x)  cos 1 ( x  2)
D) f ( x)  sin 1 ( x  2)
E) f ( x)  tan 1 ( x  2)
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Modeling Using Sinusoidal Functions
Sine and cosine functions model many real-world situations. Physical phenomenon such as
tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily
modeled by sine and cosine functions (collectively, they are called “sinusoidal functions”).
Here are some other situations that can be modeled by a sinusoidal function:




Suppose you are on a Ferris wheel at a carnival. Your height (as you are sitting in your
seat) varies sinusoidally.
Suppose you are pushing your child as s/he sits in a swing. Your child’s height varies
sinusoidally.
The motion of a swinging pendulum varies sinusoidally.
Stock prices sometimes vary sinusoidally.
We’ll work a couple of examples involving sinusoidal variation.
Recall: Given the functions f ( x)  A sin( Bx  C )  D or f ( x)  A cos( Bx  C )  D ;
2
B
The amplitude is: A
The vertical shift is: D (up if positive)
C
The horizontal shift is
(to the right if positive)
B
The period is:
Example 8: Determine the equation of the sine function which has amplitude is 5, the phase shift
is 4 to the left, the vertical shift is 3 down, and the period is 2.
20
Example 9: The number of hours of daylight in Boston is given by
 2

f ( x)  3 sin 
( x  79)   12 where x is the number of days after January 1. What is the:
 365

a. amplitude?
b. period?
c. maximum value of f(x)?
Example 10: The function P(t )  120  40 sin( 2 t ) models the blood pressure (in millimeters
of mercury) for a person who has a blood pressure of 160/90 (which is high); t represents
seconds. What is the period of this function? What is the amplitude?
21
Example 11: Determine the function of the form f ( x)  A sin( Bx ) given the following graph:
Example 12: Determine the function of the form f ( x)  A sin( Bx  C )  D given the graph:
22
Example 13: Assume that you are aboard a research submarine doing submerged training
exercises in the Pacific Ocean. At time t = 0 you start purposing (alternately deeper and then
shallower). At time t = 4 min you are at your deepest, y = – 1000 m. At time t = 9 min you next
reach your shallowest, y = –200 m. Assume that y varies sinusoidally with time. Find an
equation expressing y as a function of t.


A) f (t )  600 cos t  4   200
5



B) f (t )  400 cos t  4   600
5



C) f (t )  200 cos t  4   400
5



D) f (t )  600 cos t  4   400
5



E) A) f (t )  400 cos t  9  600
15


(Extra) Example: A signal buoy in the Gulf of Mexico bobs up and down with the height h of
its transmitter (in feet) above sea level modeled by h(t )  A sin( Bt )  5 . During a small squall its
height varies from 1 ft to 9 ft and there are 4 seconds from one 9-ft height to the next. What are
the values of the constants A and B?
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