Inverse Trigonometric Functions
Written by tutor Lauren B.
What is an Inverse Trigonometric Function?
An inverse trigonometric function is a function in which you can input a number and get/output an angle (usually in radians). It is the inverse function of the basic trigonometric functions.
Notation: The inverse function of sine is sin^{-1}(x)=arcsin(x), read as “the arcsine of x.” As a function, we can say that y=arcsin(x). This implies that sin(y)=x.
How it works: Recall from the unit circle sin(^{π}/_{4}) = ^{√2}/_{2}. We find this by looking at the y-coordinate of the point evaluated at the angle ^{π}/_{4}.
This means that if we plug in ^{√2}/_{2} into the arcsine function, we should get ^{π}/_{4} as the output. However, if we look at the unit circle, we also see that there is on other angle that has ^{√2}/_{2}
for its y-coordinate; the angle ^{3π}/_{4}. This indicates that arcsine is not a function because it is not one-to-one (one input gives two or more outputs).
Thus, we need to limit its range to [-^{π}/_{2}, ^{π}/_{2}] so it is a function (i.e., it will only give one output).
Domain/Range of Inverse Trigonometric Functions
The domain and range of the common inverse trigonometric functions is as follows:
y=sin(x) | y=cos(x) | y=tan(x) |
Domain: all real numbers
(angle) |
Domain: all real numbers
(angle) |
Domain: all real numbers except
^{nπ}/_{2} when n is odd |
Range: [-1,1] | Range: [-1,1] | Range: all real numbers |
Each of these functions demonstrates a cyclic nature (has a repeating pattern). Because of this, we need to limit the domains of sine, cosine, and tangent so that each is one to one and will have an inverse. We do this as follows:
y=sin(x) | y=cos(x) | y=tan(x) |
Domain: [-^{π}/_{2},^{π}/_{2}] | Domain: [0,π] | Domain: (-^{π}/_{2},^{π}/_{2}) |
Range: [-1,1] | Range: [-1,1] | Range: all real numbers |
Now that each function has an inverse, we see that the domain and range of each inverse trigonometric function are as follows:
y=arcsin(x) | y=arccos(x) | y=arctan(x) |
Domain: [-1,1] | Domain: [-1,1] | Domain: all real numbers |
Range: [-^{π}/_{2},^{π}/_{2}] | Range: [0,π] | Range: (-^{π}/_{2},^{π}/_{2}) |
Here are the graphs:
Evaluating Inverse Trigonometric functions
Example 1: Find arccos(^{1}/_{2}).
Solution: Keeping in mind that the range of arccosine is [0,π], we need to look for the x-values on the unit circle that are ^{1}/_{2} and on the top half of the unit circle.
We find that when the angle is ^{π}/_{3} x=^{1}/_{2}, so arccos(^{1}/_{2}) = ^{π}/_{3}.
Example 2: Find arctan(1).
Solution: We need to find an angle that has a tangent of 1. Since tan(x) = ^{sin(x)}/_{cos(x)}, we need to find an angle whose sine and cosine
are the same. The range of arctangent is (-^{π}/_{2},^{π}/_{2}); putting these two criteria together, we see that the angle must be
^{π}/_{4}. In other words, arctan(1) = ^{π}/_{4}.
Example 3: What is x if sin(x) = -1?
Solution: We can solve for x by using inverse trigonometric functions. If we take the arcsine of both sides,
sin(x) = -1
arcsin(sin(x)) = arcsin(-1)
x = arcsin(-1)
we see that sine and arcsine cancel each other. So now we need to find the angle where y = -1 on the unit circle. A quick glance shows us that the angle we need is ^{3π}/_{2}, but this is not in the range of
arcsine. Using reference angles, we find that ^{3π}/_{2} has the same values as –^{π}/_{2}, which is in the range of arcsine. So x = –^{π}/_{2}.
Using a calculator to evaluate inverse trigonometric functions
Inverse trigonometric functions can also be found using a scientific or graphing calculator. Hit the “second” or “shift” button, then the trigonometric function that you need. Make sure you are in the correct mode for what your assignment requires. Using the previous example, it should be entered into a calculator as such:
SHIFT > COS (1/2) so on the calculator it would look like this: COS^{-1}(1/2)
Is y=sin^{-1}(x) the same as y=(sin(x))^{-1}?
No. These are very different functions. y=sin^{-1}(x) is an inverse trigonometric function; whereas y=(sin(x))^{-1} is a reciprocal trigonometric function.
As an inverse function, we can simplify y=(sin(x))^{-1} = ^{1}/_{sin(x)} = csc(x); the input is an angle and the output is a number, the same as the regular sine function.
This is the same for ALL Trigonometric functions.
The graphs are very different as well:
y=arcsin(x) | y=csc(x) |
Inverse Trig Functions Quiz
True or False: An inverse trigonometric function is the same as a reciprocal trigonometric function.
True
False
What is x if x = arcsin(0)?
^{π}/_{2}
π
–^{π}/_{2}
0
Find arccos(-^{√2}/_{2}).
^{π}/_{4}
–^{π}/_{4}
^{3π}/_{4}
^{5π}/_{4}
What is x if ^{√3}/_{2} = sin(x)?
^{π}/_{6}
^{5π}/_{6}
^{7π}/_{6}
–^{π}/_{6}
Find arctan(√3).
^{2π}/_{3}
^{π}/_{3}
^{π}/_{6}
^{4π}/_{3}