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Inverse Trigonometric Functions

Author: Sophia

what's covered

In this lesson, you will use inverse trigonometric functions to find the angle that corresponds to the value of a trigonometric function. For example, suppose you are standing 100 feet away from a tree that is 40 feet tall. An inverse trigonometric function is used to determine the angle of elevation to the top of the tree. Specifically, this lesson will cover:

1. Defining the Inverse Sine, Cosine, and Tangent Functions
2. Finding Exact Values of Inverse Sine, Cosine, and Tangent Functions
3. Approximating Values of Inverse Sine, Cosine, and Tangent Functions
4. Finding Unknown Angles of Right Triangles

1. Defining the Inverse Sine, Cosine, and Tangent Functions

1a. The Inverse Sine Function

Consider the function y equals sin x shown in the figure.

A graph with an x-axis ranging from −7pi/6 to 7pi/6 and a y-axis ranging from −2 to 2, in increments of 0.5. The graph has a horizontal S-shaped curve that begins at (−7pi over 6, 0.5) and ends at (7pi over 6, −0.5). The curve passes through several marked points at (−pi, 0), (−pi over 2, −1), (0, 0), (pi over 2, 1), and (pi, 0).

Since the function is not one-to-one, its inverse is not a function.

The graph below shows the function y equals sin x on the restricted domain open square brackets short dash straight pi over 2 comma space straight pi over 2 close square brackets.

$A graph with an x-axis ranging from −5pi over 6 to 5pi over 6 and a y-axis ranging from −2 to 2, in increments of 0.5. The graph has a curve that begins in the third quadrant from the marked point at (−pi over 2, −1), increases slowly then more rapidly through (0, 0) \, then begins to increase more gradually as it comes to a stop at the point (pi over 2, 1).$

On the restricted domain, y equals sin x is one-to-one, and the range is open square brackets short dash 1 comma space 1 close square brackets comma which is the same as the unrestricted sine function.

Recall that the domain of a function is the range of its inverse function, and the range of a function is the domain of its inverse function.

Therefore, on the restricted domain, the inverse of the sine function has domain open square brackets short dash 1 comma space 1 close square brackets and range

To find the inverse of y equals sin x comma first interchange x and y to get x equals sin y. Since there is no mathematical operation that can be used to solve for y, we define the inverse sine function as follows.

The inverse sine function, or the function y equals sin to the power of short dash 1 end exponent x comma is the value of y such that sin y equals x comma where short dash 1 less or equal than x less or equal than 1 and short dash straight pi over 2 less or equal than y less or equal than straight pi over 2.

hint

The “-1” in the expression sin to the power of short dash 1 end exponent x

is NOT an exponent. A common mistake is to rewrite sin to the power of short dash 1 end exponent x

as $fraction numerator 1 over denominator sin x end fraction.$ The notation sin to the power of k x

means

open parentheses sin x close parentheses to the power of k

when

This means that the inverse sine function receives a number between -1 and 1 as input, and returns an angle between short dash straight pi over 2 and as its output.

We know that $sin open parentheses straight pi over 4 close parentheses equals fraction numerator square root of 2 over denominator 2 end fraction.$ Since is between short dash straight pi over 2 and straight pi over 2 comma it follows that $sin to the power of short dash 1 end exponent open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses equals straight pi over 4.$

try it

Consider the statement sin open parentheses short dash straight pi over 6 close parentheses equals short dash 1 half.

Write an equivalent statement using an inverse trigonometric function.

sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses equals short dash straight pi over 6

The graph of y equals sin to the power of short dash 1 end exponent x is shown below.

A graph with an x-axis ranging from −2 to 2, in increments of 0.5 and a y-axis ranging from −3pi over 4 to 3pi over 4. The graph has a curve that begins from the third quadrant at the marked point at (−1, −pi over 2), curves downward as it rises through the point (0, 0), then opens upward as it rises, then comes to a stop at the point (1, pi over 2).

term to know

Inverse Sine Function

The function y equals sin to the power of short dash 1 end exponent x

is the value of y such that sin y equals x comma

where

short dash 1 less or equal than x less or equal than 1

and

short dash straight pi over 2 less or equal than y less or equal than straight pi over 2.

1b. The Inverse Cosine Function

Consider the function y equals cos x shown in the figure.

Since the function is not one-to-one, its inverse is not a function.

The graph below shows the function y equals cos x on the restricted domain open square brackets 0 comma space straight pi close square brackets.

$A graph with an x-axis ranging from −pi/6 to 7pi/6 and a y-axis ranging from −1.5 to 1.5, in increments of 0.5. A curve begins from the marked point at (0, 1), opens downward as it decreases \through the point (pi/2, 0), then opens upward as it decreases to the point (pi, −1), where the graph ends.$

On the restricted domain, y equals cos x is one-to-one, and the range is open square brackets short dash 1 comma space 1 close square brackets comma which is the same as the unrestricted cosine function.

Therefore, its inverse function has domain open square brackets short dash 1 comma space 1 close square brackets and range open square brackets 0 comma space straight pi close square brackets.

To find the inverse of y equals cos x comma first interchange x and y to get x equals cos y. Since there is no mathematical operation that can be used to solve for y, we define the inverse cosine function as follows.

The inverse cosine function, or the function y equals cos to the power of short dash 1 end exponent x comma is the value of y such that cos y equals x comma where short dash 1 less or equal than x less or equal than 1 and 0 less or equal than y less or equal than straight pi.

try it

Consider the statement cos open parentheses 0 close parentheses equals 1.

Write an equivalent statement using an inverse trigonometric function.

cos to the power of short dash 1 end exponent open parentheses 1 close parentheses equals 0

The graph of y equals cos to the power of short dash 1 end exponent x is shown below.

term to know

Inverse Cosine Function

The function y equals cos to the power of short dash 1 end exponent x

is the value of y such that cos y equals x comma

where

and

0 less or equal than y less or equal than straight pi.

1c. The Inverse Tangent Function

Consider the function y equals tan x shown in the figure.

A graph with an x-axis ranging from −2.5 to 1.5, in increments of 0.5 and a y-axis ranging from 0 to 5pi over 4. A curve starts from a marked point at (−1, pi) in the second quadrant, decreases sharply at first then more gradually, passes through the marked point at (0, pi over 2) then decreases more gradually as it ends at the point (1, 0).

Since the function is not one-to-one, its inverse is not a function.

The graph below shows the function y equals tan x on the restricted domain open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses.

On the restricted domain, y equals tan x is one-to-one, and the range is open parentheses short dash infinity comma space infinity close parentheses comma which matches the range of the unrestricted tangent function.

Therefore, its inverse function has domain open parentheses short dash infinity comma space infinity close parentheses and range open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses.

To find the inverse of y equals tan x comma first interchange x and y to get x equals tan y. Since there is no mathematical operation that can be used to solve for y, we define the inverse tangent function as follows.

The inverse tangent function, or the function y equals tan to the power of short dash 1 end exponent x comma is the value of y such that tan y equals x comma where x is any real number and short dash straight pi over 2 less than y less than straight pi over 2.

try it

Consider the statement tan open parentheses straight pi over 4 close parentheses equals 1.

Write an equivalent statement using an inverse trigonometric function.

tan to the power of short dash 1 end exponent open parentheses 1 close parentheses equals straight pi over 4

The graph of y equals tan to the power of short dash 1 end exponent x is shown below.

did you know

There are also inverse trigonometric functions for secant, cosecant, and cotangent, but they are rarely used at this level since they are related to the inverse cosine, sine, and tangent functions, respectively.

The inverse sine, cosine, and tangent functions are also called arcsin x comma

and

The “arc” prefix is used to show that the function returns the length of the arc (in radians) for the corresponding trigonometric function and argument.

The output of an inverse trigonometric function is an angle. The input of an inverse trigonometric function is often called the argument of the function. You may recall we used this word to describe the input of a logarithmic function.

term to know

Inverse Tangent Function

The function y equals tan to the power of short dash 1 end exponent x

is the value of y such that tan y equals x comma

where x is any real number and short dash straight pi over 2 less than y less than straight pi over 2.

2. Finding Exact Values of Inverse Sine, Cosine, and Tangent Functions

When the argument of an inverse sine or cosine function is one of the special values of the corresponding trigonometric function (positive or negative), the unit circle can be used to evaluate the function exactly. A unit circle is shown below.

A fully labeled unit circle diagram. Key coordinates are shown for angles at 0, 30, 45, 60, and 90 degrees in the first quadrant, including (1, 0), (sqrt(3)/2, 1/2), (sqrt(2)/2, sqrt(2)/2), (1/2, sqrt(3)/2), and (0, 1).

EXAMPLE

Evaluate $sin to the power of short dash 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses.$

The value is the angle such that $sin theta equals fraction numerator square root of 3 over denominator 2 end fraction.$
On the first revolution of the unit circle, there are two possible answers: and $fraction numerator 2 straight pi over denominator 3 end fraction.$
is in the interval , but $fraction numerator 2 straight pi over denominator 3 end fraction$ is not in the interval

Thus, $sin to the power of short dash 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses equals straight pi over 3.$

When the argument of the inverse trigonometric function is negative (and in the domain of the function), you could still use the unit circle, but reference angles can also be used to find the value of the inverse trigonometric function, keeping in mind the range of the inverse trigonometric function.

EXAMPLE

Find the exact value of $cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 2 over denominator 2 end fraction close parentheses.$

We want to find the angle theta

so that $cos theta equals short dash fraction numerator square root of 2 over denominator 2 end fraction.$ On the first revolution of the unit circle, there are two possible answers: $fraction numerator 3 straight pi over denominator 4 end fraction$ and $fraction numerator 5 straight pi over denominator 4 end fraction.$ $fraction numerator 3 straight pi over denominator 4 end fraction$ is in the interval open square brackets 0 comma space straight pi close square brackets comma

open square brackets 0 comma space straight pi close square brackets comma

but $fraction numerator 5 straight pi over denominator 4 end fraction$ is not in the interval open square brackets 0 comma space straight pi close square brackets.

open square brackets 0 comma space straight pi close square brackets.

Thus, $cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 2 over denominator 2 end fraction close parentheses equals fraction numerator 3 straight pi over denominator 4 end fraction.$

EXAMPLE

Evaluate

sin to the power of short dash 1 end exponent open parentheses short dash 1 close parentheses.

We want to find the angle theta

so that

On the first revolution of the unit circle, there is one possible answer: $fraction numerator 3 straight pi over denominator 2 end fraction.$ Since $fraction numerator 3 straight pi over denominator 2 end fraction$ is not in the range of the inverse sine function, subtract 2 straight pi

to get an angle coterminal to $fraction numerator 3 straight pi over denominator 2 end fraction.$ The result is short dash straight pi over 2 comma

which is in the interval open square brackets short dash straight pi over 2 comma space straight pi over 2 close square brackets.

Thus,

sin to the power of short dash 1 end exponent open parentheses short dash 1 close parentheses equals short dash straight pi over 2.

try it

Consider the expression sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses.

Find the exact value of the expression.

Refer back to the special values of the tangent function. Knowing these is very helpful to evaluate the inverse tangent function.

EXAMPLE

Evaluate

tan to the power of short dash 1 end exponent open parentheses short dash 1 close parentheses.

We want to find the angle theta

so that

On the first revolution of the unit circle, there are two possible answers: $fraction numerator 3 straight pi over denominator 4 end fraction$ and $fraction numerator 7 straight pi over denominator 4 end fraction.$ Neither angle is in the interval open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses comma

open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses comma

but since $fraction numerator 7 straight pi over denominator 4 end fraction$ is in quadrant IV, subtract 2 straight pi

to get an angle coterminal to it. The result is short dash straight pi over 4 comma

which is in the interval open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses.

Thus,

tan to the power of short dash 1 end exponent open parentheses short dash 1 close parentheses equals short dash straight pi over 4.

try it

Consider the expression tan to the power of short dash 1 end exponent open parentheses square root of 3 close parentheses.

Find the exact value of this expression.

tan to the power of short dash 1 end exponent open parentheses square root of 3 close parentheses equals straight pi over 3

3. Approximating Values of Inverse Sine, Cosine, and Tangent Functions

On your calculator, usually above the buttons SIN, COS, and TAN, you should see three labels marked SIN to the power of short dash 1 end exponent comma COS to the power of short dash 1 end exponent comma and TAN to the power of short dash 1 end exponent comma which are used to evaluate trigonometric functions. They are usually 2nd functions, meaning you might need to press the “Shift” or “2nd” keys to access them.

As long as the input value is in the domain of the inverse trigonometric function that you use, the calculator will return a value in decimal form.

If your calculator is in radian mode, the inverse trigonometric functions will return an angle in radian form. Most calculators will return an approximate form, but some have the capability to return the exact form, especially if the input value is one of the special values discussed earlier.

If your calculator is in degree mode, the inverse trigonometric functions will return an angle measured in degrees.

For example, consider the expression $cos to the power of short dash 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses.$

When in radian mode, your calculator most likely will return the value 0.5236 (this value is rounded to four decimal places).

When in degree mode, your calculator will return the value 30, which means 30 degree.

On the other hand, consider the expression sin to the power of short dash 1 end exponent open parentheses 1.4 close parentheses.

In either degree or radian mode, your calculator will return an error message. Since 1.4 is not in the domain of the inverse sine function, there is no output value for an input value of 1.4.

try it

Evaluate each inverse trigonometric function in radians to four decimal places.

Evaluate tan⁻¹(4.17).

tan to the power of short dash 1 end exponent open parentheses 4.17 close parentheses almost equal to 1.3354

Evaluate cos⁻¹(-0.43).

cos to the power of short dash 1 end exponent open parentheses short dash 0.43 close parentheses almost equal to 2.0153

Evaluate sin⁻¹(0.854).

sin to the power of short dash 1 end exponent open parentheses 0.854 close parentheses almost equal to 1.0236

try it

Evaluate each inverse trigonometric function in degrees, to the nearest tenth.

Evaluate tan⁻¹(4.17).

tan to the power of short dash 1 end exponent open parentheses 4.17 close parentheses almost equal to 76.5 degree

Evaluate cos⁻¹(-0.43).

cos to the power of short dash 1 end exponent open parentheses short dash 0.43 close parentheses almost equal to 115.5 degree

Evaluate sin⁻¹(0.854).

sin to the power of short dash 1 end exponent open parentheses 0.854 close parentheses almost equal to 58.6 degree

try it

Evaluate each inverse trigonometric function in degrees.

Evaluate sin⁻¹(-2.5).

sin to the power of short dash 1 end exponent open parentheses short dash 2.5 close parentheses

is undefined.

Evaluate cos⁻¹(-2.5).

cos to the power of short dash 1 end exponent open parentheses short dash 2.5 close parentheses

is undefined.

Evaluate tan⁻¹(-2.5).

tan to the power of short dash 1 end exponent open parentheses short dash 2.5 close parentheses equals short dash 68.2 degree

4. Finding Unknown Angles of Right Triangles

Up to this point, we could only find unknown angles of a right triangle if one other angle was known. Now that we’ve learned about inverse trigonometric functions, we can find an angle associated with the ratio of two sides.

EXAMPLE

Consider the triangle below, and find the angle theta

to the nearest degree.

A right-angled triangle with the vertical side labeled ‘7’, the hypotenuse labeled ‘16’, and the horizontal side unlabeled. The right angle between the vertical and horizontal sides is represented by a small square. The angle between the hypotenuse and the base side is labeled ‘theta’.

A right-angled triangle with the vertical side labeled ‘7’, the hypotenuse labeled ‘16’, and the horizontal side unlabeled. The right angle between the vertical and horizontal sides is represented by a small square. The angle between the hypotenuse and the base side is labeled ‘theta’.

	Referencing the angle its opposite side and the hypotenuse are known. Therefore, the sine function should be used to relate the sides and the angle.
	means
	Approximate the value using a calculator, in degree mode.

try it

Consider the right triangle shown here.

A right-angled triangle with the horizontal side labeled ‘15’, the vertical side labeled ‘8’, and the hypotenuse unlabeled. The right angle between the vertical and horizontal sides is represented by a small square. The angle labeled ‘A’ is marked between the hypotenuse and the horizontal side.

A right-angled triangle with the horizontal side labeled ‘15’, the vertical side labeled ‘8’, and the hypotenuse unlabeled. The right angle between the vertical and horizontal sides is represented by a small square. The angle labeled ‘A’ is marked between the hypotenuse and the horizontal side.

Find the measure of angle A to the nearest degree.

tan to the power of short dash 1 end exponent open parentheses 8 over 15 close parentheses almost equal to 28 degree

These ideas can also be applied to real-world problems.

watch

In the following video, we’ll find the angle of inclination for a hot air balloon that is rising.

EXAMPLE

A truss for the roof of a house is constructed using two identical right triangles. Each has a base of 12 feet and height 4 feet, as shown in the figure.

A truss is composed of two right triangles side by side with a common vertical height of 4 feet. Each has a base of 12ft, and the angle between the hypotenuse and the base of each triangle is labeled ‘theta’ and the angle between the height and base is represented by a small square, indicating that this is a right angle.

A truss is composed of two right triangles side by side with a common vertical height of 4 feet. Each has a base of 12ft, and the angle between the hypotenuse and the base of each triangle is labeled ‘theta’ and the angle between the height and base is represented by a small square, indicating that this is a right angle.

Find the angle of elevation from the base of the truss to the top of the center. Round to the nearest degree.

	Let be the angle adjacent to the side with length 12 feet. Using this angle as a reference, the sides opposite and adjacent to the angle are known, which means tangent is the best choice to relate the sides and the angle.
	Reduce means
	Approximate the angle using a calculator.

Thus, the angle of elevation from the bottom of the truss to the top of the center is about 18 degree.

try it

A 20-foot ladder is leaning against the side of a building, and reaches a height of 16 feet along the building.

What angle, in degrees, does the ladder make with the building? Round to the nearest whole degree.

theta equals cos to the power of short dash 1 end exponent open parentheses 16 over 20 close parentheses almost equal to 37 degree

summary

In this lesson, you learned how to define the inverse sine, cosine, and tangent functions. These inverse trigonometric functions are used to find the angle that corresponds to the value of a trigonometric function. You also learned that when evaluating inverse trigonometric functions, you can find the exact values of inverse sine, cosine, and tangent functions if the input value corresponds to a special angle. You can also find the approximate values of inverse sine, cosine, and tangent functions by using a calculator. Finally, you explored a few examples of real-world scenarios that demonstrate how inverse trigonometric functions are very useful for finding unknown angles of right triangles.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know

Inverse Cosine Function: The function $y equals cos to the power of short dash 1 end exponent x$ is the value of y such that $cos y equals x comma$ where $short dash 1 less or equal than x less or equal than 1$ and $0 less or equal than y less or equal than straight pi.$
Inverse Sine Function: The function $y equals sin to the power of short dash 1 end exponent x$ is the value of y such that $sin y equals x comma$ where $short dash 1 less or equal than x less or equal than 1$ and $short dash straight pi over 2 less or equal than y less or equal than straight pi over 2.$
Inverse Tangent Function: The function $y equals tan to the power of short dash 1 end exponent x$ is the value of y such that $tan y equals x comma$ where x is any real number and $short dash straight pi over 2 less than y less than straight pi over 2.$