# Chapter 8: Right Triangle Trigonometry

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

## Introduction

Chapter 8 explores right triangles in far more depth than Chapters 4 and 5. Recall that a right triangle is a triangle with exactly one right angle. In this chapter, we will first prove the Pythagorean Theorem and its converse, followed by analyzing the sides of certain types of triangles. Then, we will introduce trigonometry, which starts with the tangent, sine and cosine ratios. Finally, we will extend sine and cosine to any triangle, through the Law of Sines and the Law of Cosines.

## Chapter Outline

- 8.1. Pythagorean Theorem and Pythagorean Triples
- 8.2. Applications of the Pythagorean Theorem
- 8.3. Inscribed Similar Triangles
- 8.4. 45-45-90 Right Triangles
- 8.5. 30-60-90 Right Triangles
- 8.6. Sine, Cosine, Tangent
- 8.7. Trigonometric Ratios with a Calculator
- 8.8. Trigonometry Word Problems
- 8.9. Inverse Trigonometric Ratios
- 8.10. Laws of Sines and Cosines

### Chapter Summary

## Summary

This chapter begins with the Pythagorean Theorem, its converse, and Pythagorean triples. Applications of the Pythagorean Theorem are explored including finding heights of isosceles triangles, proving the distance formula, and determining whether a triangle is right, acute, or obtuse. The chapter then branches out into special right triangles, 45-45-90 and 30-60-90. Trigonometric ratios, trigonometry word problems, inverse trigonometric ratios, and the Law of Sines and Law of Cosines are explored at the end of this chapter.

### Chapter Keywords

- Pythagorean Theorem
- Pythagorean Triple
- Distance Formula
- Pythagorean Theorem Converse
- Geometric Mean
- 45-45-90 Corollary
- 30-60-90 Corollary
- Trigonometry
- Adjacent (Leg)
- Opposite (Leg)
- Sine Ratio
- Cosine Ratio
- Tangent Ratio
- Angle of Depression
- Angle of Elevation
- Inverse Tangent
- Inverse Sine
- Inverse Cosine
- Law of Sines
- Law of Cosines

### Chapter Review

Solve the following right triangles using the Pythagorean Theorem, the trigonometric ratios, and the inverse trigonometric ratios. When possible, simplify the radical. If not, round all decimal answers to the nearest tenth.

Determine if the following lengths make an acute, right, or obtuse triangle. If they make a right triangle, determine if the lengths are a Pythagorean triple.

- 11, 12, 13
- 16, 30, 34
- 20, 25, 42
- \begin{align*}10 \sqrt{6}, 30, 10 \sqrt{15}\end{align*}
- 22, 25, 31
- 47, 27, 35

Find the value of \begin{align*}x\end{align*}.

- The angle of elevation from the base of a mountain to its peak is \begin{align*}76^\circ\end{align*}. If its height is 2500 feet, what is the distance a person would climb to reach the top? Round your answer to the nearest tenth.
- Taylor is taking an aerial tour of San Francisco in a helicopter. He spots AT&T Park (baseball stadium) at a horizontal distance of 850 feet and down (vertical) 475 feet. What is the angle of depression from the helicopter to the park? Round your answer to the nearest tenth.

Use the Law of Sines and Cosines to solve the following triangles. Round your answers to the nearest tenth.

### Texas Instruments Resources

*In the CK-12 Texas Instruments Geometry FlexBook, 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/9693.*