On these restricted domains, we can define the
inverse trigonometric functions .
The
inverse sine function
means
The inverse sine function is sometimes called the
arcsine function, and notated
The
inverse cosine function
means
The inverse cosine function is sometimes called the
arccosine function, and notated
The
inverse tangent function
means
The inverse tangent function is sometimes called the
arctangent function, and notated
The graphs of the inverse functions are shown in
[link] ,
[link] , and
[link] . Notice that the output of each of these inverse functions is a
number, an angle in radian measure. We see that
has domain
and range
has domain
and range
and
has domain of all real numbers and range
To find the
domain and
range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line
Relations for inverse sine, cosine, and tangent functions
For angles in the interval
if
then
For angles in the interval
if
then
For angles in the interval
if
then
Writing a relation for an inverse function
Given
write a relation involving the inverse sine.
Finding the exact value of expressions involving the inverse sine, cosine, and tangent functions
Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically
(30°),
(45°), and
(60°), and their reflections into other quadrants.
Given a “special” input value, evaluate an inverse trigonometric function.
Find angle
for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
If
is not in the defined range of the inverse, find another angle
that is in the defined range and has the same sine, cosine, or tangent as
depending on which corresponds to the given inverse function.
Evaluating inverse trigonometric functions for special input values
Evaluate each of the following.
Evaluating
is the same as determining the angle that would have a sine value of
In other words, what angle
would satisfy
There are multiple values that would satisfy this relationship, such as
and
but we know we need the angle in the interval
so the answer will be
Remember that the inverse is a function, so for each input, we will get exactly one output.
To evaluate
we know that
and
both have a sine value of
but neither is in the interval
For that, we need the negative angle coterminal with
To evaluate
we are looking for an angle in the interval
with a cosine value of
The angle that satisfies this is
Evaluating
we are looking for an angle in the interval
with a tangent value of 1. The correct angle is