Section 3.4 – Inverse Trigonometric Functions
Learning Objectives
Welcome to Section 3.4! In this section, you will…
- Understand and use the inverse sine, cosine, and tangent functions.
- Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
- Find exact values of composite functions with inverse trigonometric functions.
For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.
Understanding and Using the Inverse Sine, Cosine, and Tangent Functions
In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1.

For example, if
- Since
, then . - Since
, then . - Since
, then .
In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if
Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0. Figure 2 shows the graph of the sine function limited to

Figure 3 shows the graph of the tangent function limited to

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
- The inverse sine function
means . The arcsine function, and notated .
- The inverse cosine function
means . The arccosine function, and notated .
- The inverse tangent function
means . The arctangent function, and notated .
The graphs of the inverse functions are shown in Figure 4, Figure 5, and Figure 6. Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that



Relations for Inverse Sine, Cosine, and Tangent Functions
For angles in the interval
For angles in the interval
For angles in the interval
Example 1
Writing a Relation for an Inverse Function
Given
Show/Hide Solution
Use the relation for the inverse sine. If
In this problem,
Try It #1
Given
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
How To
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.
Example 2
Evaluating Inverse Trigonometric Functions for Special Input Values
Evaluate each of the following.
ⓐ
ⓑ
ⓒ
ⓓ
Show/Hide Solution
ⓐ Evaluating
ⓑ To evaluate
ⓒTo evaluate
ⓓ Evaluating
Try It #2
Evaluate each of the following.
ⓐ
ⓑ
ⓒ
ⓓ
How To
Given two sides of a right triangle like the one shown in Figure 7, find an angle.

- If one given side is the hypotenuse of length
and the side of length adjacent to the desired angle is given, use the equation . - If one given side is the hypotenuse of length
and the side of length opposite to the desired angle is given, use the equation . - If the two legs (the sides adjacent to the right angle) are given, then use the equation
.
Example 3
Applying the Inverse Cosine to a Right Triangle
Solve the triangle in Figure 8 for the angle

Show/Hide Solution
Because we know the hypotenuse and the side adjacent to the angle, it makes sense for us to use the cosine function.
Finding Exact Values of Composite Functions with Inverse Trigonometric Functions
There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let
Evaluating Compositions of the Form
For any trigonometric function,
Compositions of a trigonometric function and its inverse
Q&A
Is it correct that
No. This equation is correct if
How To
Given an expression of the form f−1(f(θ)) where
- If
is in the restricted domain of , then . - If not, then find an angle
within the restricted domain of such that . Then .
Example 4
Using Inverse Trigonometric Functions
Evaluate the following:
ⓐ
ⓑ
ⓒ
ⓓ
Show/Hide Solution
ⓐ
ⓑ
ⓒ
ⓓ
Try It #5
Evaluate
Evaluating Compositions of the Form
Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form

Because
How To
Given functions of the form
- If
is in , then . - If x is not in
, then find another angle y in such that .
- If
is in , then . - If
is not in , then find another angle in such that .
Example 5
Evaluating the Composition of an Inverse Sine with a Cosine
Evaluate
ⓐby direct evaluation.
ⓑ by the method described previously.
Show/Hide Solution
ⓐ Here, we can directly evaluate the inside of the composition.
Now, we can evaluate the inverse function as we did earlier.
ⓑ We have
Try It #6
Evaluate
Evaluating Compositions of the Form
To evaluate compositions of the form
Example 6
Evaluating the Composition of a Sine with an Inverse Cosine
Find an exact value for
Show/Hide Solution
Beginning with the inside, we can say there is some angle such that
Since

We know that the inverse cosine always gives an angle on the interval
Try It #7
Evaluate
Example 7
Evaluating the Composition of a Sine with an Inverse Tangent
Find an exact value for
Show/Hide Solution
While we could use a similar technique as in Example 5, we will demonstrate a different technique here. From the inside, we know there is an angle such that

Using the Pythagorean Theorem, we can find the hypotenuse of this triangle.
Now, we can evaluate the sine of the angle as the opposite side divided by the hypotenuse.
This gives us our desired composition.
Try It #8
Evaluate
Example 8
Finding the Cosine of the Inverse Sine of an Algebraic Expression
Find a simplified expression for
Show/Hide Solution
We know there is an angle
Because we know that the inverse sine must give an angle on the interval
Try It #9
Find a simplified expression for
Media
Access this online resource for additional instruction and practice with inverse trigonometric functions.
3.4 Section Exercises
Verbal
1. Why do the functions
2. Since the functions
3. Explain the meaning of
4. Most calculators do not have a key to evaluate
5. Why must the domain of the sine function,
6. Discuss why this statement is incorrect:
7. Determine whether the following statement is true or false and explain your answer:
Algebraic
For the following exercises, evaluate the expressions.
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For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.
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For the following exercises, find the angle
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For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.
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For the following exercises, find the exact value of the expression in terms of
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Extensions
For the following exercises, evaluate the expression without using a calculator. Give the exact value.
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For the following exercises, find the function if
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Graphical
48. Graph
49. Graph
50. Graph one cycle of
51. For what value of
52. For what value of
Real-World Applications
53. Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?
54. Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?
55. An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.
56. Without using a calculator, approximate the value of
57. A truss (interior beam structure) for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.
58. The line
59. The line
60. What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example a 5% grade means that the road rises 5 feet for every 100 feet of horizontal distance.)
61. A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder's angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?
62. Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?