# 2.4: Inverse Functions and Trigonometric Equations

**At Grade**Created by: CK-12

## Inverse Trigonometric Functions

*General Definitions of Inverse Trigonometric Functions*

**In-Text Examples**

1) Students who don’t refer to the illustration may get mixed up about which angle is

Students may also be using the wrong key on their calculators to find the inverse tangent; they may be just using the “tan” key by itself, or they may be using a combination of the “tan” key and the reciprocal

**Review Questions**

1) Some students will still be confusing the vertical and horizontal line tests here, or otherwise be fuzzy on the idea of what makes a relation a function. (Also, the inverse of relation

2) Many students will try to make side

**Additional Problems**

1) While walking home, you decide to take a shortcut across an empty lot. From one corner of the lot, you cannot see the opposite corner, but you know the lot is

**Answers to Additional Problems**

1)

*Using the “Inverse” Notation*

**In-Text Examples**

1) Students may need a refresher on the unit circle at this point; in particular, they may have forgotten how to figure out what quadrant the reference triangle should be in.

**Review Questions**

1) This is a trick question; students may get confused by the fact that

## Exact Values of Inverse Functions

*Exact Values of Special Inverse Circular Functions*

**Review Questions**

Using special triangles may be the wrong approach for these problems, as the angles here are not acute ones and therefore can’t be directly found in triangles. Triangles may still be useful for figuring out the reference angles for the angles given, but that’s after students figure out what quadrant the angles are in.

When using the unit circle, though, some students may need to be reminded that they can simply read off the sine and cosine values from the coordinates of the points on the circle, as the sine values is equal to the

*Range of the Outside Function, Domain of the Inside Function*

**In-Text Examples**

1) Students may not realize yet that the easiest way to prove two functions are each other’s inverses is to compose the two functions and prove that the composition always equals

**Review Questions**

1) Both of these functions are actually not invertible simply because their inverses are not functions.

*Applications, Technological Tools*

**Additional Problems**

1) Find the inverse of the following function:

**Answers to Additional Problems**

1)

## Properties of Inverse Circular Functions

*Derive Properties of Other Five Inverse Circular Functions in Terms of Arctan*

**In-Text Examples**

1) Remind students to label which angle is

(Of course, they can simply plug the values of

2) On part a, students may miss the part about the bottom of the screen not being directly on the ground. On part b, they may be confused by the fact that they don’t have enough information to find exact angle measures. Instead, they must find a formula for the angle measure in terms of

**Review Questions**

2) After all the previous discussion of how

**Additional Problems**

1) Express the function

**Answers to Additional Problems**

1)

*Derive Inverse Cofunction Properties*

**Review Questions**

Remembering to get the domain restrictions right should be the only tricky part here.

*Find Exact Values of Functions of Inverse Functions Using Pythagorean Triples*

**In-Text Examples**

3) The most likely error here is getting the quadrant wrong.

## Applications of Inverse Circular Functions

*Revisiting*

**In-Text Examples**

Students may make the same errors here that they made when they originally covered this topic, particularly getting the various shifts and stretches mixed up and getting the sign of the phase shift backwards.

**Additional Problems**

1) How would you express the equation from problem 1 above as a transformation of

**Answers to Additional Problems**

1)

*Solving for Particular Values in Trigonometric Equations*

**Review Questions**

1) Students may still be unsure how to derive an equation from just the two points given. The trick is to realize that the horizontal distance between the two points is half the period (and the frequency is

**Additional Problems**

1) You are riding a Ferris wheel at an amusement park.

**Answers to Additional Problems**

1) The equation that models this problem is

Using a calculator gives us an answer of approximately

## Trigonometric Equations

*Solving Trigonometric Equations Analytically*

**In-Text Examples**

1) Students may get stuck on these if they don’t remember the basic identities.

2) Make sure to point out on part b that there are a total of four solutions on the given interval.

**Review Questions**

1) Of course, students are likely to miss the fact that they need to double the interval of possible solutions (as explained in the solution key). Even if they do figure this out, they may not realize that this means the two solutions within the interval

2) As on other problems, students may forget to look for solutions in more than one quadrant. They may also forget to consider the cosines of both

3) The trick here is to treat

4) The same errors as on problem 1 apply.

*Solve Trig Equations (Factoring)*

**In-Text Examples**

2) Students will of course be tempted to divide both sides of the equation by tan

**Review Questions**

1) Upon finding that

2) Again, beware of dividing through by

*Solve Equations (Using Identities)*

**Review Questions**

1) Getting the signs reversed when factoring is the most likely error here.

2) Students are likely to forget the “for all values of

## Trigonometric Equations with Multiple Angles

*Solve Equations (with Double Angles)*

**In-Text Examples**

1) Again, beware of dividing both sides by

4) …or by

**Review Questions**

1) Finding the actual measure of angle

2) The solution to part b involves realizing that

3) This equation is actually true for all values of

*Solving Trigonometric Equations Using Half Angle Formulas*

**In-Text Examples**

3) Students may be momentarily stumped by the appearance of the unfamiliar-looking angle

**Review Questions**

3) Students will get three solutions to this equation if they follow the procedure outlined in the solution key, but not all of those solutions will in fact be correct. Squaring both sides of an equation introduces extraneous solutions, so they must check their answers afterward to see which ones need to be discarded. This is true any time both sides of an algebraic equation are squared.

*Solving Trigonometric Equations with Multiple Angles*

**In-Text Examples**

2) Once again, remind students that they must look for values of

**Review Questions**

2) Students may get stuck trying to figure out how to express

3) After finding the principal solutions to the equation, students may try to derive the other solutions by adding

## Equations with Inverse Circular Functions

*Solving Trigonometric Equations Using Inverse Notation*

**In-Text Examples**

2) Students may be momentarily confused by the lack of “arc”-function buttons on their calculators, and may need to be reminded to use the

**Review Questions**

3-4) Students are likely to stop when they have found the one solution within the restricted range, and forget that they are looking for solutions between

*Solving Trigonometric Equations Using Inverse Functions*

**In-Text Examples**

3) The most likely error here is for students to try “unwrapping” the right-hand side of the equation in the wrong order, for example by taking the inverse cosine before dividing by

**Review Questions**

1) This problem is subject to the same error as example 3 above; also, students who do not recognize that the problem relies on an angle sum identity will approach it wrong from the beginning and get an answer that makes no sense, if they get any answer at all.

*Solving Inverse Equations Using Trigonometric Identities*

**In-Text Examples**

1) This example is much trickier than it seems at first, because of the double angle. Watch out for students forgetting to account for the

**Review Questions**

1) Students may get their double- and half-angle identities mixed up here, or may forget that if they are applying a double-angle identity to

2) (Note:

Students may think they don’t have enough information to solve this problem, but they do if they express the legs of the triangles as

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