Chapter 8: Right Triangle Trigonometry
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.
 8.1.
Pythagorean Theorem and Pythagorean Triples
 8.2.
Applications of the Pythagorean Theorem
 8.3.
Inscribed Similar Triangles
 8.4.
454590 Right Triangles
 8.5.
306090 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, 454590 and 306090. 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
 454590 Corollary
 306090 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

106√,30,1015−−√  22, 25, 31
 47, 27, 35
Find the value of
 The angle of elevation from the base of a mountain to its peak is
76∘ . 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 CK12 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.