On these restricted domains, we can define the
inverse trigonometric functions .
The
inverse sine functiony=sin−1x means
x=siny. The inverse sine function is sometimes called the
arcsine function, and notated
arcsinx.
y=sin−1xhas domain[−1,1]and range[−π2,π2]
The
inverse cosine functiony=cos−1x means
x=cosy. The inverse cosine function is sometimes called the
arccosine function, and notated
arccosx.
y=cos−1xhas domain[−1,1]and range[0,π]
The
inverse tangent functiony=tan−1x means
x=tany. The inverse tangent function is sometimes called the
arctangent function, and notated
arctanx.
y=tan−1xhas domain(−∞,∞)and range(−π2,π2)
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
sin−1x has domain
[−1,1] and range
[−π2,π2],cos−1x has domain
[−1,1] and range
[0,π], and
tan−1x has domain of all real numbers and range
(−π2,π2). 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
y=x.
The sine function and inverse sine (or arcsine) functionThe cosine function and inverse cosine (or arccosine) functionThe tangent function and inverse tangent (or arctangent) function
Relations for inverse sine, cosine, and tangent functions
For angles in the interval
[−π2,π2], if
siny=x, then
sin−1x=y.
For angles in the interval
[0,π], if
cosy=x, then
cos−1x=y.
For angles in the interval
(−π2,π2), if
tany=x, then
tan−1x=y.
Writing a relation for an inverse function
Given
sin(5π12)≈0.96593, write a relation involving the inverse sine.
Use the relation for the inverse sine. If
siny=x, then
sin−1x=y .
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
π6 (30°),
π4 (45°), and
π3 (60°), and their reflections into other quadrants.
Given a “special” input value, evaluate an inverse trigonometric function.
Find angle
x for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
If
x is not in the defined range of the inverse, find another angle
y that is in the defined range and has the same sine, cosine, or tangent as
x, depending on which corresponds to the given inverse function.
Evaluating inverse trigonometric functions for special input values
Evaluate each of the following.
sin−1(12)
sin−1(−√22)
cos−1(−√32)
tan−1(1)
Evaluating
sin−1(12) is the same as determining the angle that would have a sine value of
12. In other words, what angle
x would satisfy
sin(x)=12? There are multiple values that would satisfy this relationship, such as
π6 and
5π6, but we know we need the angle in the interval
[−π2,π2], so the answer will be
sin−1(12)=π6. Remember that the inverse is a function, so for each input, we will get exactly one output.
To evaluate
sin−1(−√22), we know that
5π4 and
7π4 both have a sine value of
−√22, but neither is in the interval
[−π2,π2]. For that, we need the negative angle coterminal with
7π4:sin−1(−√22)=−π4.
To evaluate
cos−1(−√32), we are looking for an angle in the interval
[0,π] with a cosine value of
−√32. The angle that satisfies this is
cos−1(−√32)=5π6.
Evaluating
tan−1(1), we are looking for an angle in the interval
(−π2,π2) with a tangent value of 1. The correct angle is
tan−1(1)=π4.