4.6
Inverse Trigonometric Functions
With many of the previous elementary functions, we able to create inverse functions. For example:
the inverse of a linear function is another linear function;
the inverse of the quadratic function is (with restricted domain) the square root function;
the inverse of an exponential function is a logarithm.
And so on....
4.6.1
The inverse of a trig function
Here we examine inverse functions for the six basic trig functions.
Recall that if we are going to take a function f (x) and create the inverse function f −1 (x) then the
function f (x) needs to be one-to-one. We cannot have two different inputs a and b where y = f (a) = f (b)
for then we don’t know how to compute f −1 (y). Visually, this says that the graph of y = f (x) must pass
the horizontal line test.
This is a significant problem for the trig functions since all the trig functions are periodic and so,
given any y-value, there are an infinite number of x-values such that y = f (x). Trig functions badly fail
the horizontal line test! We will fix this problem (in the next section) by appropriately restricting the
domain of the trig functions in order to create inverse functions.
Before we go deeper into describing inverse trig functions, let us take a moment to review the inverse
function concept. In the past we used the superscript −1 to indicate an inverse function, writing f −1 (x)
to mean the inverse function of f (x). We will continue to do this, writing sin−1 x for the inverse sine
function and tan−1 x for the inverse function of tangent. Etc. But there is another common notation for
inverse functions in trigonometry. It is common to write “arc ” to indicate an inverse function, since the
output of an inverse function is the angle (arc) which goes with the trig value. For example, the inverse
function of sin(x) is written either sin−1 (x) or arcsin(x). In these notes these terms are equivalent.
Before we go into detail on the inverse trig functions, let’s practice the concept of an inverse function.
A Worked Problem
1. Find (without a calculator) the exact values of the following:
√
2
2 )
√
arccos(− 23 )
√
arcsin( 23 )
(a) arccos(
(b)
(c)
(d) arctan(1)
√
(e) arctan( 3)
√
(f) arctan(− 3)
Solutions.
(a) Since
arccos(x) is the
inverse function
of cos(x) then we seek here an angle θ whose cosine is
√
√
√
2
2
2
π
π
2 . Since cos( 4 ) = 2 then arccos( 2 ) should be 4 .
√
(b) arccos(−
√
(c) arcsin(
3
2 )
3
2 )
=
=
π
3
5π
6
since cos( 5π
6 )=−
since sin( π3 ) =
√
√
3
2 .
3
2 .
(d) arctan(1) = π4 since tan( π4 ) = 1.
√
√
(e) arctan( 3) = π3 since tan( π3 ) = 3.
√
√
(f) arctan(− 3) = − π3 since tan(− π3 ) = − 3.
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4.6.2
Restricting the range of trig functions to create inverse functions
Since the trig functions are periodic there are an infinite number of x-values such that y = f (x). We can
fix this problem by restricting the domain of the trig functions so that the trig function is one-to-one in
that specific domain.
For example, the sine function has domain (−∞, ∞) and range [−1, 1]. It is periodic with period 2π
and during each period, each output occurs twice. Figure 24, below, gives a graph of the sine function on
the interval from [−10, 10]. Note that an output such as y = 21 has many, many inputs associated with it.
Figure 24. The sine function on the interval [−10, 10].
We can make the sine function one-to-one if we restrict the domain to a region (of length π) in which
each output occurs exactly once. If we restrict the domain of sine to [− π2 , π2 ] then suddenly the previous
graph (in Figure 24) looks like this (in Figure 25).
Figure 25. The sine function defined just on the interval [− π2 , π2 ]
This new restricted sine function is one-to-one; it satisfies the horizontal line test! Here (in Figure 26,
below) is the same function with the scale on the x-axis drawn out a bit more.
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Figure 26. The sine function on the interval [− π2 , π2 ], zooming in on the x-axis.
Now we are ready to create an inverse function. The restricted function drawn above has domain [− π2 , π2 ]
and range [−1, 1]. If we swap the inputs and outputs we have a new function with domain [−1, 1] and
range [− π2 , π2 ]. Here is the graph of the arcsine.
Figure 27. The arcsine function
(In may not be immediately obvious but the graph in Figure 27 is not the graph in Figure 26. It is
instead the graph in Figure 26 after we have swapped the x- and y-axes. The graph is Figure 27 is what
we get by taking the graph in Figure 26 and reflecting it across the line y = x.
4.6.3
The arcsine and arcosine functions
The inverse function for sin(x) is called “inverse sine” or “arc sine.” (I will try to use the term “arcsine”,
written arcsin(x).) The arcsine function has domain [−1, 1] and range [− π2 , π2 ]. It has the property that
on the interval [−1, 1], if y = arcsin(x) then x = sin(y).
As the sine function takes in an angle and outputs a real number between −1 and 1, the arcsin takes
in a value between −1 and 1 and gives out a corresponding angle. For example, arcsin( 21 ) must be the
angle π6 since sin( π6 ) = 12 .
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The Arccosine function
Unfortunately the restricted domain choice we made for the sine function doesn’t work as the restricted
domain for cosine since cosine is not one-to-one on the interval [− π2 , π2 ]; also cosine is nonnegative on
this interval and we want to choose a domain that represents all of the range [−1, 1]. For cosine we will
instead choose the restricted domain [0, π] when we look for the inverse function.
Here is a graph of cosine on the interval [−10, 10].
Figure 28. The cosine function, failing the horizontal line test.
Now we restrict the domain to the region [0, π] so that each output occurs exactly once.
Figure 29. The cosine function on the interval [0, π]
If we exchange x (inputs) with y (outputs) and so reflect that graph across the line y = x we get the
graph of the arccosine function below.
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Figure 30. The arccosine function
4.6.4
The other inverse trig functions
In the creation of inverse trig functions, we restrict the domain of the original trig function to an interval
of length π. This means we will choose our angles to fall into two quadrants of the unit circle. We choose
those quadrants with the following properties:
1. We always include the first quadrant ([0, π2 ]) in our domain.
2. The other quadrant is adjacent to the first quadrant, so it is either Quadrant II or Quadrant IV.
3. We need to make sure that all values of output (including negative values) are included in the
range, so this means the “other” quadrant of the domain is Quadrant II for cosine and secant and
Quadrant IV for sine and cosecant.
The Arctangent function
The tangent function, like all trig functions, is periodic. It has period π.
Figure 31. The tangent function
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We restrict the domain of tan x to an interval of period π so that tangent hits each output exactly once.
We can do that if we restrict the tangent to [− π2 , π2 ]. (For the tangent function we will include Quadrant
IV so that we don’t cross a place (such as π2 or − π2 ) where the tangent is undefined.)
In Figure 32, below, we hide (in light yellow) the other branches of the tangent function and focus on
the interval [− π2 , π2 ] where the tangent function is one-to-one.
Figure 32. The tangent function on the interval [− π2 , π2 ]
Now we are ready to create the inverse function (below, in Figure 33.)
Figure 33. The arctangent function
Arcsecant and Arccosecant
There are similar definitions for the restricted domains that allow us to invert the functions sec(x) and
csc(x). We restrict sec(x) to the same domain as its reciprocal, cos(x), and we restrict csc(x) to the same
domain as its reciprocal, sin(x). We have to be a little careful here since sec(x) is undefined at x = π/2
since cos(π/2) is zero. So technically the new domain of sec(x) is not [0, π] but [0, π/2) ∪ (π/2, π]. (We
won’t spend much time worrying about these details as long as we understand the mechanism of inverse
functions.)
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4.6.5
Applied Problems with Inverse Trig Functions
A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it touches level ground 360
feet from the base of the tower. What angle does the wire make with the ground? Express your answer
using degree measure rounded to one decimal place.4
Solution.
The wire forms the hypotenuse of a right triangle in which the right angle is at the base of the
tower. The 360 feet from the base of the tower to the spot where the guy wire touches the ground forms
360
another side of the triangle with the angle θ between those two sides. So cos θ = 1000
= 0.36. Therefore
◦
θ = arccos(0.36) ≈ 1.20253 ≈ 68.9 .
A security camera is to be installed in a jewelry store. It is to be installed 20 feet from the center
of a jewelry counter which is 30 feet long. We want the camera to rotate back and forth so as to cover
the entire jewelry counter. Through what angle should the camera rotate so as to just cover the entire
counter. (Give the answer to the nearest degree.)5
Solution.
The camera needs to rotate 15 feet in both directions from the center of the counter. The camera,
the center of the counter and an end of the counter form a right triangle (with right angle at the
15
center of the counter) which has short sides of length 20 feet and 15 feet. Let θ = arctan( 20
) ≈
◦
0.6435 radians ≈ 36.87 . This is the angle the camera must cover from the center of the counter to
one end. So 2θ ≈ 1.287 radians ≈ 73.74◦ is the total angle the camera must rotate since we want the
camera to cover the counter from one end to the other.
4.6.6
Other resources on inverse trig functions
(Hard copy references need to be fixed.) In the free textbook, Precalculus, by Stitz and Zeager (version
3, July 2011, available at stitz-zeager.com) this material is covered in section 10.6.
In the free textbook, Precalculus, An Investigation of Functions, by Lippman and Rassmussen (Edition
1.3, available at www.opentextbookstore.com) this material is covered in section 6.3.
In the textbook by Ratti & McWaters, Precalculus, A Unit Circle Approach, 2nd ed., c. 2014 this
material appears in section ??.
In the textbook by Stewart, Precalculus, Mathematics for Calculus, 6th ed., c. 2012 (here at Amazon.com)
this material appears ??.
Here are some online resources:
1. Khan Academy videos on inverse trig functions
2. Dr. Paul’s online math notes include a review of inverse trig functions.
Homework.
As class homework, please complete Worksheet 4.6, The Inverse Trig Functions available
through the class webpage.
4 This
5 This
is problem 212 on page 846 of the free textbook, Precalculus, by Stitz and Zeager (version 3, July 2011.)
a modification of a problem ocurring in the first edition of the textbook by Ratti & McWaters, p. 322.
183