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Math 170 - Cooley
Pre-Calculus
OCC
Section 5.5 – The Inverse Trigonometric Functions
We will now construct inverse functions from our six trigonometric functions by restricting the domain of the
each function so that it will be one-to-one. We need each graph of a function and its inverse to be able to pass
the Vertical Line Test (to be a function) and the Horizontal Line Test (to be one-to-one which means
invertible.) See section 1.8.
y = sin x
y = cos x
  x  
  x  
y = sin x


y = cos x

2
x
2
y = sin–1 x
0 x 
y = cos–1 x
y = tan x
  x   , except for
odd multiples of 2
y = tan x



2
x
2
y = tan–1 x
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Math 170 - Cooley
Pre-Calculus
OCC
Section 5.5 – The Inverse Trigonometric Functions
Inverse Function
Name
Usual
Notation
Definition
Domain
Range
Inverse Sine
(arcsine)
y  sin 1 x
y = arcsin x
x = sin y
[–1, +1]
  
 2 , 2 


Inverse Cosine
(arccosine)
y  cos 1 x
y = arcos x
x = cos y
[–1, +1]
0,  
Inverse Tangent
(arctangent)
y  tan 1 x
y = arctan x
x = tan y
(–, +)
  
 2 , 2 


Inverse Function
Name
Usual
Notation
Inverse Cosecant
(arccosecant)
y  csc 1 x
y = arccsc x
y  sec 1 x
Inverse Secant
(arcsecant)
y = arcsec x
Inverse Cotangent
(arccotangent)
Definition
Domain
x = csc y
(–, –1] 
[+1, +)
x = sec y
(–, –1] 
[+1, +)
x = cot y
(–, +)
1
y  cot x
y = arccot x
Range


2
 y

2
, y0
0  y , y 

2
0,  
Properties of Inverse Functions
sin 1  sin x   x where 

x

cos1 (cos x)  x
2
2
1
sin  sin x   x where 1  x  1
cos  cos x   x
1
tan 1  tan x   x
tan  tan x   x
1
where 

x
where 0  x  
where 1  x  1

2
2
where   x  
Notation for Inverse Functions
The inverse functions are denoted by a superscript “–1” between the name of the function and its argument.
This notation does not mean an exponent of –1, but is similar to inverse function notation such as f 1 read “f
inverse”. So, sin 1 x is read “sine inverse of x” , and …
sin x 2  sin 2 x , but
1
sin 1 x  sin x  .
(Note:  sin x  
1
1
 csc x , so obviously sin 1 x  csc x or we wouldn’t have different names for them!)
sin x
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Math 170 - Cooley
Pre-Calculus
OCC
Section 5.5 – The Inverse Trigonometric Functions
1

 Example: Find the exact value of the expression: sin  cos1 
2

Solution 
So, we need to find the angle  , 0     , whose cosine equals
cos 

1
. Thus,
2
1
, 0 
2

3
1
3

 
Now, sin  cos1   sin   
2

 3 2
 Example: Use a calculator to find the value of the following expression rounded to two decimal places.
csc 1 (5)
Solution 


1
1
, since csc1 (5)  sin 1   .
2
2
5
 5
1
1
Now, sin   , so  lies in quadrant I. The calculator yields sin 1    0.20 , which is an
5
 5
So, we seek to find the angle  , 
 
, whose sine equals
angle in quadrant I, so csc1 (5)  0.20
 Example: Use a calculator to find the value of the following expression rounded to two decimal places.
cot 1 ( 5)
Solution 
So, we seek to find the angle  , 0     , whose tangent equals 
1
, since
5
1
 1 
cot 1 (  5)  tan 1  
 . Now, tan    5 , so  lies in quadrant II. The calculator yields
5

 1 
tan 1  
  0.42 , which is an angle in quadrant IV. Since,  is in quadrant II,
5

  0.42    2.72 . Therefore, cot 1 (  5)  2.72 .
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Math 170 - Cooley
Pre-Calculus
OCC
Section 5.5 – The Inverse Trigonometric Functions
 Exercises:
Find the exact value of each expression without using a calculator.
1) cos 1 1
2) arccos(1)
3) tan 1 (1)

3
4) sin 1  

 2 
 3
5) tan 1 

 3 
 2 
6) csc 1 

 3
7) sec1 (1)
8) arccsc(1)
9) cot 1 1
10) cot 1  0 
Use a calculator to find the value of each expression rounded to two decimal places.
11) cos 1 0.6
12) sin 1 (0.23)
13) tan 1 (0.4)
14) cos1 (0.44)
15) sec1 ( 3)
16) cot 1 ( 8.1)
Find the exact value, if any, of each expression. Do not use a calculator.
   
17) sin 1 sin    
  10  

 2 
18) cos cos 1    
 3 

  3
19) sin 1 sin  
  7
21) tan  tan 1  2  
  7  
22) cos 1 cos 

  4 
23) cos1 cos  



20) sin sin 1  2  
24) cos cos 1   
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Math 170 - Cooley
Pre-Calculus
OCC
Section 5.5 – The Inverse Trigonometric Functions
 Exercises: Find the exact value of each expression.

 1 
25) tan sin 1    
 2 



3 
26) cos sin 1  

 2  

27) sec tan 1 3
2 

28) tan 1  tan

3 


 2 
29) cos sin 1 
 3  

 

30) sin  tan 1  3 

 1 
31) csc  tan 1   
 2 

32) sec1 (2)
33) csc1
 Answers:
1) 0; 2)  ; 3) 

4
; 4) 

3
; 5)

6
; 6)

3
; 7) 0 ; 8) 


2
; 9)

4
; 10)

2

 2
; 11) 0.93;
2
3
; 18)  ; 19) 
;
3
10
7
3
1


20) Not defined; 21) –2; 22)
; 23)  ; 24) Not defined; 25) 
; 26)
; 27) 2; 28)  ;
3
4
3
2
7
3 10
3
2

29)
; 30) 
; 31) 
; 32)
; 33)
.
3
10
3
3
4
12) 0.288  0.23 ; 13) –0.38; 14) 2.03; 15) 1.91; 16) 3.02; 17) 

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