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Chapter 2
Section 2.5
Inverse Trigonometric Functions
Inverse Trigonometric Functions
In this section we focus on how to get the value for the angle (either in degrees or
radians) if you know the value for the sine, cosine or tangent. This process is
formalized in what are called the inverse trigonometric functions. For some
numbers the values of these functions can be found by hand, but for most a
calculator is needed.
Inverse Sine Function
This function is commonly denoted in one of two ways: sin-1x
or
arcsin(x)
The meaning of this function is as follows for any number x between -1 and 1:
sin-1x = arcsin(x) = The angle between -/2 and /2 whose sine is x.
Think of the “answer” to the inverse sine function as an angle  that has a certain
relationship with x. The angle  will have the following property:
IF  = sin-1x = arcsin(x) THEN sin = x
For example, sin-1(½) = arcsin(½) is the number =/6, because sin(/6) = ½. The
tricky part here is that the “answer” to the inverse sine function must be between
the angles -/2 and /2 (-90 and 90 if you are using degrees).

2
The restrictions you have on the values  and x
for the relation  = sin-1x = arcsin(x) are:
x  sin 
1. x the input must be between -1 and 1

1
2. The angle  the output is between -/2 and /2
Both of these come from the values of sine on
the unit circle. (Remember the sine is the
distance up and down on the unit circle.)

2
Notice that x can not get bigger than 1 or less than -1 and hit the unit circle with a
horizontal line. The value of sin-1x is undefined if x is bigger than 1 or less than -1.
The horizontal green line hits the circle in 2 places once on the right half and on
the left half. Since the angles on the left half are not between -/2 and /2 we do
not use them for the inverse sine.
To find the values for the inverse sine for certain numbers we can do using this
idea of drawing the unit circle and going down by the appropriate amount. The
numbers that you can find the values for the inverse sine by hand are:
1
2
3
0,  1,  , 
,
2
2
2
Find:
sin 1
 
 3
2
Find: arcsin
 
2
2
Find:
sin 1  1
2
2
 60
 3
2
1
2
sin 1
   60
 3
2

 3
2
2

4
arcsin
1
   45
2
2
 90

0
 4
sin 1  1  90 

2
For numbers other than the ones that represent sides of either a 45-45-90, a
30-60-90 triangle or 0 or ±1 you will need to use a calculator.
Example: Find sin-1(.75) with a calculator
sin-1(.75) = .848062…
(if calculator is in radian mode)
sin-1(.75) = 48.5904…
(if calculator is in degree mode)
Inverse Cosine Function
This function is commonly denoted in one of two ways: cos-1x
or
arccos(x)
The meaning of this function is as follows for any number x between -1 and 1:
cos-1x = arccos(x) = The angle between 0 and  whose cosine is x.
Think of the “answer” to the inverse cosine function as an angle  that has a
certain relationship with x. The angle  will have the following property:
IF  = cos-1x = arccos(x) THEN cos = x
For example, cos-1(½) = arccos(½) is the number =/3, because cos(/3) = ½.
Unlike the inverse sine the “answer” to the inverse cosine function must be
between the angles 0 and  (0 and 180 if you are using degrees).
x  cos
The restrictions you have on the values  and x
for the relation  = cos-1x = arccos(x) are:
1. x the input must be between -1 and 1
2. The angle  the output is between 0 and 
Both of these come from the values of cosine on
the unit circle. (Remember the cosine is the
distance left and right on the unit circle.)


1
0
Find:
cos 1  21 
For numbers other than the ones that represent sides of
either a 45-45-90, a 30-60-90 triangle or 0 or ±1
you will need to use a calculator.
1
2
Example: Find cos-1(.375) with a calculator
3
2
120
cos-1(.375) = 1.1864…

(if calculator is in radian mode)
cos-1(.375) = 67.9757…
cos 1  21   120 
2
3
(if calculator is in degree mode)
Inverse Tangent Function
This function is commonly denoted in one of two ways: tan-1x
or
arctan(x)
The meaning of this function is as follows for any number x between -1 and 1:
tan-1x = arctan(x) = The angle between -/2 and /2 whose tangent is x.
Think of the “answer” to the inverse cosine function as an angle  that has a
certain relationship with x. The angle  will have the following property:
IF  = tan-1x = arctan(x) THEN tan = x
For example, tan-1(1) = arctan(1) is the number  =/4, because tan(/4) = 1. Just
like the inverse sine the “answer” to the inverse tangent function must be between
the angles -/2 and /2 (-90 and 90 if you are using degrees).
Algebraic and Numeric Properties of Inverse Trigonometric Functions
The inverse trigonometric functions have relationships with all the other
trigonometric functions. Rather than listing out some more trigonometric identities
(I know you must be getting tired of that at this point) I want to show you a method
for getting them.
Construction of Corresponding Right Triangle
What you do is to construct a triangle that conforms to the inverse trig relation you
have. This is something we did before when we were doing right triangle trig.
Suppose  = cos-1(⅔) = arcos(⅔). We make a right triangle with one angle  and
label the sides so that cos = ⅔. We then solve for the remaining side.
2 b 3
2
4  b2  9
3
5
2
b 5
2

2
b 5
2


coscos   
tan cos   
sin cos 1 2 3  
1 2
1 2
3
3
5
3
2
3
5
2
cot cos 1 2 3  


csccos   
sec cos 1 2 3  
1 2
3
The values of all the trig functions for  !
2
5
3
2
3
5
The construction of the corresponding right
triangle will even work for algebraic
expressions. For this it is often useful to think
of the number x as x/1.
  tan 1 x
-1
Example: Change cos(tan x) to an
x  tan 
expression with x.
x
1
x2 1
1


This can also be done for the other trigonometric functions.


x
x 2 1




cot tan 1 x 
tan tan 1 x  x
c  x2 1

cos tan 1 x 
2. Solve for the remaining side.
sin tan 1 x 
x2 1  c2
x
 tan 
1. Set  =
and construct the
corresponding triangle by labeling the sides.
tan-1(x)
x 2  12  c 2

1
x 2 1

sec tan 1 x 
1
x
x 2 1
1
There are several identities that are very useful.



sin sin 1 x  x
sin 1 sin x   x

2
 x  2



cos cos 1 x  x
tan tan 1 x  x
cos 1 cos x   x 0  x  
tan 1 tan x   x

2
 x  2