Section 6.1 – Inverse Trigonometric Functions
Objective: In this section you will learn how to evaluate the inverse trigonometric functions and compositions of
trigonometric functions and inverse trigonometric functions.
I.
Inverse Sine Function
The sine function is not a one-to-one function. We know this because the graph of y  sin x fails the
_______________________________________. Therefore, to define an inverse function for sine we must
restrict the domain. The domain of y  sin x is restricted to ______________________________ in order to
have a one-to-one function that obtains all possible values of y  sin x .
The inverse sine function is defined by:
y  sin 1 x if and only if _____________________ where _____  x  _____ and _____  y  ______.
That is, the domain of y  arcsin x is ___________________ and the range is ______________________.
Example 1: Find the exact value:
a. arcsin( 1) = ______________
c.
II.
b.
sin 1 (1/ 2) = _______________
arcsin(  3 / 2) = _____________
Inverse Cosine Function
The inverse cosine function is defined by:
y  cos 1 x if and only if _____________________ where _____  x  _____ and _____  y  ______.
That is, the domain of y  arccos x is ___________________ and the range is ______________________.
Example 2: Find the exact value:
a.
arccos(1 / 2) = ______________
c.
arccos( ) = _______________
b.
cos 1 ( 2 / 2) = __________
III.
Inverse Tangent Function
The inverse tangent function is defined by:
y  tan 1 x if and only if _____________________ where _____  x  _____ and _____  y  ______.
That is, the domain of
y  arctan x is ___________________ and the range is ______________________.
Example 3: Find the exact value:
a.
arctan( 3 ) = ___________
tan 1 (1) = _______________
b.
Example 4: Use a calculator to approximate to four decimal places.
IV.
a.
arccos( 0.85) = ________________
c.
arctan( 8) = __________________
b.
sin 1 (4.215) = ________________
Other Inverse Trigonometric Functions
I will not expect you to memorize the domains and ranges for the other inverse trigonometric functions. You may
refer to the chart on p.265 of your text to assist you in evaluating these functions. I do expect you to memorize
the domain and range for the inverse functions of sine, cosine, and tangent.
V.
Compositions of Functions
Example 5:
a.
1

sin  sin 1  = _________________
2

c.
sin sin 1  = __________________



tan tan 1 1 = _______________


sin sin 1 x = _____________.
Similarly, as long as ___________________________,
a.

 2
 = ___________________
sin  arcsin

2



Hence, as long as ___________________________,
Example 6:
b.


cos cos1 x = _____________.
b.


tan arctan 3 = ____________________
Hence, as long as ____________________________, tan(arctan x ) = __________________.
Example 7:
Hence,
a.
 
sin 1  sin  = _________________
4

c.
 11 
sin 1  sin
 = ________________
6 

b.
2 

arcsin  sin
 = ___________________
3 

sin 1 sin x   x only when ____________________________. If x is not in this interval, BE CAREFUL!
Similarly,
arctan tan x  x only when __________________________.
Example 8:
a.


arctan  tan  = _______________
3

b. tan  tan
Example 9:
a.


cos 1  cos  = ______________
3

b. arccos cos
Hence,
1




3 
 = _________________
4 
7 
 = _________________
6 
cos 1 cos x   x only when ____________________________. If x is not in this interval, BE CAREFUL!
What if you need to find the composition of a trig function and the inverse of a different trig function?
Example 10:
a.
3

sin  cos 1  = _________________
7

b.
3
5

cos sin 1  cos 1  = _________________
4
13 
