# Chapter 4: Inverse Trigonometric Functions

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

- 4.1.
## Basic Inverse Trigonometric Functions

- 4.2.
## Graphing Inverse Trigonometric Functions

- 4.3.
## Inverse Trigonometric Properties

- 4.4.
## Applications & Models

### Chapter Summary

In this chapter, we studied all aspects of inverse trigonometric functions. First, we defined the function by finding inverses algebraically. Second, we analyzed the graphs of inverse functions. We needed to restrict the domain of the trigonometric functions in order to take the inverse of each of them. This is because they are periodic and did not pass the horizontal line test. Then, we learned about the properties of the inverse functions, mostly composing a trig function and an inverse. Finally, we applied the principles of inverse trig functions to real-life situations.

## Chapter Vocabulary

- Arccosecant
- Read “cosecant inverse” and also written . The domain of this function is all reals, excluding the interval (-1, 1). The range is all reals in the interval .

- Arccosine
- Read “cosine inverse” and also written . The domain of this function is [-1, 1]. The range is .

- Arccotangent
- Read “cotangent inverse” and also written . The domain of this function is all reals. The range is .

- Arcsecant
- Read “secant inverse” and also written . The domain of this function is all reals, excluding the interval (-1, 1). The range is all reals in the interval .

- Arcsine
- Read “sine inverse” and also written . The domain of this function is [-1, 1]. The range is .

- Arctangent
- Read “tangent inverse” and also written . The domain of this function is all reals. The range is .

- Composite Function
- The final result from when one function is plugged into another, .

- Harmonic Motion
- A motion that is consistent and periodic, in a sinusoidal pattern. The general equation is where is the amplitude, is the frequency, and is the phase shift.

- Horizontal Line Test
- The test applied to a function to see if it has an inverse. Continually draw horizontal lines across the function and if a horizontal line touches the function more than once, it does not have an inverse.

- Inverse Function
- Two functions that are symmetric over the line .

- Inverse Reflection Principle
- The points and in the coordinate plane are symmetric with respect to the line . The points and are reflections of each other across the line .

- Invertible
- If a function has an inverse, it is invertible.

- One-to-One Function
- A function, where, for every value, there is EXACTLY one value. These are the only invertible functions.

## Review Questions

- Find the exact value of the following expressions:
- Use your calculator to find the value of each of the following expressions:
- Find the exact value of the following expressions:
- Find the inverse of each of the following:
- Sketch a graph of each of the following:
- Using the triangles from Section 4.3, find the following:
- A ship leaves port and travels due west 20 nautical miles, then changes course to and travels 65 more nautical miles. Find the bearing to the port of departure.
- Using the formula from Example 1 in Section 4.4, determine the measurement of the sun’s angle of inclination for a building located at a latitude of on the of May.
- Find the inverse of . HINT: Set and and rewrite and in terms of sine.
- Find the inverse of . HINT: Set and and rewrite and in terms of sine.

## Texas Instruments Resources

*In the CK-12 Texas Instruments Trigonometry FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9702.*

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