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Composite Functions Involving Inverse Trigonometric Functions

Author: Sophia
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what's covered
In this lesson, you will evaluate expressions that apply trigonometric and inverse trigonometric functions. Specifically, this lesson will cover:

Table of Contents

1. The Inverse Properties of Inverse Trigonometric Functions

Since sin   x and sin to the power of short dash 1 end exponent x are inverses of each other, it stands to reason that sin open parentheses sin to the power of short dash 1 end exponent x close parentheses equals x and sin to the power of short dash 1 end exponent open parentheses sin   x close parentheses equals x. Since sin to the power of short dash 1 end exponent x has a restricted domain and range, this is not true for all real numbers x.

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

Since the domain of sin to the power of short dash 1 end exponent x is open square brackets short dash 1 comma space 1 close square brackets comma sin open parentheses sin to the power of short dash 1 end exponent x close parentheses equals x when short dash 1 less or equal than x less or equal than 1.

Now consider the expression sin to the power of short dash 1 end exponent open parentheses sin   x close parentheses.

Since the restricted domain open square brackets short dash straight pi over 2 comma space straight pi over 2 close square brackets is considered for sin   x comma sin to the power of short dash 1 end exponent open parentheses sin   x close parentheses equals x when short dash straight pi over 2 less or equal than x less or equal than straight pi over 2.

EXAMPLE

Consider the expression sin to the power of negative 1 end exponent open parentheses sin straight pi over 3 close parentheses. Since straight pi over 3 is in the interval open square brackets short dash straight pi over 2 comma straight pi over 2 close square brackets, it follows that sin to the power of negative 1 end exponent open parentheses sin straight pi over 3 close parentheses equals straight pi over 3.

Now consider the expression sin to the power of negative 1 end exponent open parentheses sin fraction numerator 2 straight pi over denominator 3 end fraction close parentheses. Since fraction numerator 2 straight pi over denominator 3 end fraction is not in the interval open square brackets short dash straight pi over 2 comma straight pi over 2 close square brackets, it follows that sin to the power of negative 1 end exponent open parentheses sin fraction numerator 2 straight pi over denominator 3 end fraction close parentheses not equal to fraction numerator 2 straight pi over denominator 3 end fraction. We will explore how to evaluate expressions like this later.

The table below summarizes the inverse properties for the trigonometric functions.

Inverse Property Domain
sin open parentheses sin to the power of short dash 1 end exponent x close parentheses equals x open square brackets short dash 1 comma space 1 close square brackets
sin to the power of short dash 1 end exponent open parentheses sin   x close parentheses equals x open square brackets short dash straight pi over 2 comma space straight pi over 2 close square brackets
cos open parentheses cos to the power of short dash 1 end exponent x close parentheses equals x open square brackets short dash 1 comma space 1 close square brackets
cos to the power of short dash 1 end exponent open parentheses cos   x close parentheses equals x open square brackets 0 comma space straight pi close square brackets
tan open parentheses tan to the power of short dash 1 end exponent x close parentheses equals x open parentheses short dash infinity comma space infinity close parentheses
tan to the power of short dash 1 end exponent open parentheses tan   x close parentheses equals x open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses

EXAMPLE

Consider the expressions tan open parentheses tan to the power of short dash 1 end exponent 10 close parentheses and tan to the power of short dash 1 end exponent open parentheses tan   10 close parentheses.

Since tan open parentheses tan to the power of short dash 1 end exponent x close parentheses equals x for all real numbers, tan open parentheses tan to the power of short dash 1 end exponent 10 close parentheses equals 10.

Since tan to the power of short dash 1 end exponent open parentheses tan   x close parentheses equals x only in the domain open parentheses short dash straight pi over 2 comma space straight pi over 2 close parentheses comma the value of tan to the power of short dash 1 end exponent open parentheses tan   10 close parentheses is not 10.

By using a calculator, tan to the power of short dash 1 end exponent open parentheses tan   10 close parentheses almost equal to 0.5752.

Note: if you use your calculator to evaluate tan open parentheses tan to the power of short dash 1 end exponent 10 close parentheses comma the answer is 10.

EXAMPLE

Consider the expressions sin open parentheses sin to the power of short dash 1 end exponent straight pi close parentheses and sin to the power of short dash 1 end exponent open parentheses sin   straight pi close parentheses.

Since sin to the power of short dash 1 end exponent x is defined only on the interval open square brackets short dash 1 comma space 1 close square brackets comma sin to the power of short dash 1 end exponent straight pi is undefined since straight pi almost equal to 3.14.

Since sin   straight pi equals 0 comma sin to the power of short dash 1 end exponent open parentheses sin   straight pi close parentheses equals sin to the power of short dash 1 end exponent open parentheses 0 close parentheses equals 0.

try it
Consider the expression cos to the power of short dash 1 end exponent open parentheses cos open parentheses short dash straight pi over 6 close parentheses close parentheses.
Without using a calculator, evaluate the expression.
First, cos open parentheses short dash straight pi over 6 close parentheses equals fraction numerator square root of 3 over denominator 2 end fraction.

Then, cos to the power of short dash 1 end exponent open parentheses cos open parentheses short dash straight pi over 6 close parentheses close parentheses equals 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 equals straight pi over 6.

2. Evaluating Compositions of Trigonometric Functions and Inverse Trigonometric Functions

Now we’ll evaluate expressions such as cos open parentheses sin to the power of short dash 1 end exponent x close parentheses and tan open parentheses cos to the power of short dash 1 end exponent x close parentheses, where the inverse trigonometric function and the trigonometric function are not inverses.

2a. Evaluating Compositions With Special Angles and Trigonometric Values

EXAMPLE

Find the exact value of tan open parentheses sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses close parentheses without using a calculator.

  • The value of sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses is the angle theta such that sin theta equals short dash 1 half and short dash straight pi over 2 less or equal than theta less or equal than straight pi over 2.
  • By examining the unit circle, 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.
Now, evaluate tan open parentheses sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses close parentheses.

tan open parentheses sin to the power of short dash 1 end exponent open parentheses short dash 1 half close parentheses close parentheses equals tan open parentheses short dash straight pi over 6 close parentheses 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
equals short dash tan open parentheses straight pi over 6 close parentheses Use an even/odd identity.
equals short dash fraction numerator square root of 3 over denominator 3 end fraction This is the tangent function of a special angle.

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

try it
Consider the expression cos open parentheses sin 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 close parentheses.
Find the exact value of the expression.
First, note that sin 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 short dash straight pi over 4.

Then, cos open parentheses short dash straight pi over 4 close parentheses equals fraction numerator square root of 2 over denominator 2 end fraction.

Thus, cos open parentheses sin 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 close parentheses equals fraction numerator square root of 2 over denominator 2 end fraction.

2b. Evaluating Compositions Using Identities

When the trigonometric values are not from special angles, it is not efficient to find the angle first as we did with special angles. Either identities or right triangles need to be used.

EXAMPLE

Evaluate the expression cos open parentheses sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses close parentheses.

Remember that the expression sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses represents an angle. Let theta equals sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses. Then, sin theta equals 2 over 3.

Next, the expression cos open parentheses sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses close parentheses becomes cos theta when sin theta equals 2 over 3.

To find this value exactly, we should use identities or right triangles rather than the calculator.

Since theta equals sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses comma we know that theta terminates in quadrant I since the argument is positive.

Since theta is in quadrant I, we also know that cos theta greater than 0.

To find the value of cos theta comma we can use identities as we have in the past, or we can use a right triangle argument.

Using identities:

sin squared theta plus cos squared theta equals 1 This Pythagorean identity relates sin theta and cos theta.
open parentheses 2 over 3 close parentheses squared plus cos squared theta equals 1 Substitute sin theta equals 2 over 3.
4 over 9 plus cos squared theta equals 1 Simplify.
cos squared theta equals 5 over 9 Subtract 4 over 9 from both sides to isolate cos squared theta on one side.
cos theta equals square root of 5 over 9 end root Apply the square root principle. Since cos theta greater than 0 comma only the positive solution is considered.
cos theta equals fraction numerator square root of 5 over denominator 3 end fraction Simplify the radical.

Since the goal was to find the value of cos theta comma it follows that cos open parentheses sin to the power of short dash 1 end exponent open parentheses 2 over 3 close parentheses close parentheses equals fraction numerator square root of 5 over denominator 3 end fraction.

Using a right triangle:

Consider this right triangle, considering that sin theta equals 2 over 3.

A right-angled triangle with the vertical side labeled ‘2’, the hypotenuse labeled ‘3’, and the horizontal side unlabeled. An angle labeled ‘theta’ is marked between the hypotenuse and the horizontal side.

Since cos theta equals adjacent over hypotenuse comma find the length of the adjacent side. Let x equals the adjacent side.

x squared plus 2 squared equals 3 squared This is the Pythagorean theorem.
x squared plus 4 equals 9 Simplify.
x squared equals 5 Subtract 4 from both sides to isolate x squared on one side.
x equals square root of 5 Apply the square root principle. Since x is the length of a side, only the positive solution is considered.

Next, cos theta equals adjacent over hypotenuse equals fraction numerator square root of 5 over denominator 3 end fraction comma as found using the other method.

Thus, using the right triangle argument, cos theta equals fraction numerator square root of 5 over denominator 3 end fraction.

try it
Consider the expression cos open parentheses tan to the power of short dash 1 end exponent open parentheses 1 fourth close parentheses close parentheses.
Find the exact value of the expression.
First, let theta equals tan to the power of short dash 1 end exponent open parentheses 1 fourth close parentheses. Then, tan   theta equals 1 fourth comma and we are looking for the value of cos   theta.

Now, consider this triangle:

A triangle with horizontal side length 4, vertical side length 1, and slanted side length ‘x’. The angle between the horizontal and slanted sides is labeled ‘theta’.

Let x equals the unknown side. By the Pythagorean theorem, we know 4 squared plus 1 squared equals x squared comma which has the solution x equals square root of 17.

Then, cos open parentheses tan to the power of short dash 1 end exponent open parentheses 1 fourth close parentheses close parentheses equals cos   theta equals fraction numerator 4 over denominator square root of 17 end fraction.

Then, rationalize the denominator by multiplying by fraction numerator square root of 17 over denominator square root of 17 end fraction.

The final answer is fraction numerator 4 square root of 17 over denominator 17 end fraction.

EXAMPLE

Evaluate the expression tan open parentheses cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses close parentheses.

Remember that the expression cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses represents an angle. Let theta equals cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses. Then, cos theta equals short dash fraction numerator square root of 15 over denominator 8 end fraction.

Next, the expression tan open parentheses cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses close parentheses becomes tan theta when cos theta equals short dash fraction numerator square root of 15 over denominator 8 end fraction.

To find this value exactly, we’ll use identities.

Since cos theta equals short dash fraction numerator square root of 15 over denominator 8 end fraction comma we know that theta terminates in quadrant II since the argument is negative and the range of the inverse cosine function is open square brackets 0 comma space straight pi close square brackets.

Since theta is in quadrant II, we also know that tan theta less than 0.

To find the value of tan theta comma remember that tan theta equals fraction numerator sin theta over denominator cos theta end fraction. This means we can use the identity sin squared theta plus cos squared theta equals 1 to find sin theta comma which in turn can be used to find tan theta.

Note: since theta terminates in quadrant II, sin theta greater than 0.

sin squared theta plus cos squared theta equals 1 This Pythagorean identity relates sin theta and cos theta. The value of sin theta will be used to find the value of tan theta.
sin squared theta plus open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses squared equals 1 Substitute cos theta equals short dash fraction numerator square root of 15 over denominator 8 end fraction.
sin squared theta plus 15 over 64 equals 1 Simplify.
sin squared theta equals 49 over 64 Subtract 15 over 64 from both sides to isolate sin squared theta on one side.
sin theta equals square root of 49 over 64 end root Apply the square root principle. Since sin theta greater than 0 comma only the positive solution is considered.
sin theta equals 7 over 8 Simplify the radical.

Thus, sin theta equals 7 over 8.

Now, find the value of tan theta.

tan theta equals fraction numerator sin theta over denominator cos theta end fraction This is the quotient identity.
equals fraction numerator open parentheses begin display style 7 over 8 end style close parentheses over denominator open parentheses short dash begin display style fraction numerator square root of 15 over denominator 8 end fraction end style close parentheses end fraction Replace sin theta and cos theta with their values.
equals short dash fraction numerator 7 over denominator square root of 15 end fraction Simplify.
equals short dash fraction numerator 7 over denominator square root of 15 end fraction times fraction numerator square root of 15 over denominator square root of 15 end fraction equals short dash fraction numerator 7 square root of 15 over denominator 15 end fraction Rationalize the denominator.

Thus, tan open parentheses cos to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 15 over denominator 8 end fraction close parentheses close parentheses equals short dash fraction numerator 7 square root of 15 over denominator 15 end fraction.

try it
Consider the expression cos open parentheses tan to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 7 over denominator 3 end fraction close parentheses close parentheses.
Find the exact value of this expression.
First, let theta equals tan to the power of short dash 1 end exponent open parentheses short dash fraction numerator square root of 7 over denominator 3 end fraction close parentheses. Then, tan   theta equals short dash fraction numerator square root of 7 over denominator 3 end fraction comma and we are looking for cos   theta.

Given tan   theta equals short dash fraction numerator square root of 7 over denominator 3 end fraction comma this means that theta terminates in quadrant IV. This also means that cos   theta is positive.

Consider the triangle below, which shows the location of the angle theta in the 4th quadrant along with its reference angle theta apostrophe comma and the sides of the triangle:

An x and y axis intersect at the origin (0, 0). A triangle is drawn with vertices (0, 0), (3, 0), and (3, negative square root of 7). The horizontal side has length 3 and the vertical side has length square root of 7. A circular arc begins on the positive x-axis and continues counterclockwise until reaching the slanted side of the triangle in the fourth quadrant. The arc is labelled with the angle ‘theta’. The angle between the slanted side of the triangle and the positive x-axis is labeled ‘theta prime’.

Let h equals the length of the hypotenuse.

By the Pythagorean theorem, we know 3 squared plus open parentheses square root of 7 close parentheses squared equals h squared comma which means h equals 4.

Since cos   theta equals adjacent over hypotenuse comma it follows that cos   theta equals 3 over 4. This is the final answer.

It is also possible to apply these methods to expressions with variables in them. This is an important skill in calculus.

watch
In this video, we will express sin open parentheses arctan open parentheses x over 2 close parentheses close parentheses as an algebraic function of x.

summary
In this lesson, you began by summarizing the inverse properties of inverse trigonometric functions. You learned that compositions of trigonometric and inverse trigonometric functions are evaluated using a variety of methods. When the functions are inverses (for example, sine and inverse sine), then the inverse properties can be used. However, when they are not inverses of each other, the composition can be evaluated using trigonometric values of special angles or by using identities, or even by using a right triangle argument.

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.

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