Name________________________________
Date__________________________
We know that each function has an inverse relation and that this inverse relation is a
function only if the original function is one-to-one. Even though the 6 basic trig
functions fail the horizontal line test for one-to-oneness, we still want to study their
inverse behavior. We do this by restricting the domain of the original function to an
interval on which it is one-to-one, then find the inverse of the restricted function. (We
did this when looking at the square root function which is the inverse of the function
y = x2).
  
Restricting the domain of y = sin x to the interval  ,  , the restricted function
 2 2
becomes one-to-one. The inverse sine function y = sin-1x is the inverse of this restricted
portion of the sine function.
y = sin x
  
WINDOW:  ,  by [-1.2, 1.2]
 2 2
y = sin-1 x
  
WINDOW: [-1.5, 1.5] by  , 
 2 2
By the usual inverse relationship, the statements
y = sin-1x and x = sin y
  
are equivalent for y-values in the restricted domain  ,  and x-values in [-1, 1].
 2 2


This means that sin-1x can be thought of as the angle between  and whose sine is x.
2
2
Since angles and directed arcs on the unit circle have the same measure, the angle sin-1x
is also called the arcsine of x.
Inverse Sine Function (Arcsine Function)
  
The unique angle y in the interval  ,  such that sin y = x is the inverse sine (or
 2 2
-1
arcsine) of x, denoted sin x or arcsin x. The domain of y = sin-1x is [-1, 1] and the
  
range is  , 
 2 2
It helps to think of the range of y = sin-1x as being along the right-hand side of the unit
 
circle, which is traced out as angles range from  to .
2 2
1. Find the exact value of each expression without a calculator:

3
1

a. sin-1  
b. sin-1  

2
2


   
d. sin-1  sin   
  9 
 
c. sin-1  
2
  5
e. sin-1  sin 
  6

 

If you restrict the domain of y = cos x to he interval 0,  , as shown below, the restricted
function is one-to-one. The inverse cosine function y = cos-1x is the inverse of this
restricted portion of the cosine function.
y = cos x
WINDOW: 0,   by [-1, 1]
y = cos-1x
WINDOW: [-1, 1] by 0,  
By the usual inverse relationship, the statements
y = cos-1x
and
x = cos y
are equivalent for y-values in the restricted domain 0,   and the x-values in [-1, 1].
This means that cos-1 x can be thought of as the angle between 0 and  whose cosine is
x. The angle cos-1x is also the arccosine of x.
Inverse Cosine Function (Arccosine Function)
The unique angle y in the interval 0,   such that cos y = x in the inverse cosine (or
arccosine) of x denoted cos-1x or arcos x. The domain of y = cos-1x is [-1, 1] and the
range is 0,  
It helps to think of the range of y = cos-1x as being along the top half of the unit circle,
which is traced out as angles range from 0 to  .
  
If you restrict the domain of y = tan x to the interval   ,  as shown below, the
 2 2
restricted function is one-to-one. The inverse tangent function y = tan-1 x is the inverse
of this restricted portion of the tangent function.
y = tan x
y = tan-1x
  
WINDOW:  ,  by [-2, 2]
 2 2
  
WINDOW: [-4, 4] by  , 
 2 2
By the usual inverse relationship, the statements
y = tan-1x and x = tan y
  
are equivalent for y-values in the restricted domain   ,  and x-values in  , .
 2 2


This means that tan-1x can be thought of as the angle between  and whose tangent
2
2
-1
is x. The angle tan x is also the arctangent of x.
Inverse Tangent Function (Arctangent Function)
  
The unique angle y in the interval   ,  such than tan y = x is the inverse tangent
 2 2
-1
(or arctangent) of x denoted tan x or arctan x. The domain of y = tan-1x is  , 
  
and the range is   ,  .
 2 2
It helps to think of the range of y = tan-1x as being along the right-hand side of the unit
circle (minus the top and bottom points), which is traced out as angles range from


 and (noninclusive).
2
2
2. Find the exact value of the expression without a calculator.

a. cos 

-1  
2

2 
  7
c. cos-1  cos
  4
b. tan-1 3

 

3. Describe the end behavior of the function y = tan-1x.
Composing Trig and Inverse Trig Function
We have already seen the need for caution when applying the Inverse Composition Rule
to the trig functions and their inverses (example 1e and 2c). The following equations are
always true whenever they are defined:
sin (
)=x
cos(
)=x
tan(
)=x
On the other hand, the following equations are only true for x values in the “restricted”
domains of sin, cos, and tan:
sin-1(
)=x
cos-1(
)=x
tan-1(
)=x
An even more interesting phenomenon occurs when we compose inverse trig function of
one kind with trig functions of another kind, as in sin(tan-1x). Surprisingly, these trig
compositions reduce to algebraic functions that involve no trig at all. This curious
situation has profound implications in calculus, where is it is sometimes useful to
decompose nontrig functions into trig components that seem to come out of nowhere.
Try the exploration below:
Exploration: Funding Inverse Trig Functions of Trig Functions
In the right triangle shown to the right, the angle  is measured in radians.
1. Find tan  .
2. Find tan-1x.
x

3. Find the hypotenuse of the triangle as a function of x.
4. Find sin(tan-1(x)) as a ratio involving no trig functions.
5. Find sec(tan-1(x)) as a ratio involving no trig functions.
6. If x < 0, then tan-1x is a negative angle in the fourth quadrant. Verify that your
answers to parts 4 and 5 are still valid in this case.
4. Compose each of the 6 basic trig functions with sin-1x and reduce the composite
function to an algebraic expression involving no trig functions.
1
Name________________________________
Date__________________________
Find the exact value with out using a calculator.
 3

1. sin 1 

2


2. tan 1 0
1
3. cos 1  
2
4. tan 1 1
 1 
5. sin 1  

2

6. cos 1 0
7. Describe the end behavior of the function y = tan-1(x2)
Find the exact value without a calculator.

 1 
8. cos sin 1   
 2 

   
9. sin 1  cos  
  4 

 1 
10. cos 2 sin 1   
 2 

   
11. arcsin  cos  
  3 
12. Analyze the function f(x) = tan-1x for domain, range, continuity, increasing or
decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior.
Use transformations to describe how the graph of the function is related to a basic inverse
trigonometric graph. State the domain and range.
13. f(x) = sin-1(2x)
14. h(x) = 5 tan-1(x/2)
Find an exact solution to the equation without a calculator.
15. sin (sin-1 x) = 1
16. 2 sin-1 x = 1
Find an algebraic expression equivalent to the given expression (HINT: form a right
triangle as done in example 4)
17. sin(tan-1 x)
18. cos(arctan 2x)