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CHAPTER 10 Geometry © 2010 Pearson Prentice Hall. All rights reserved. 10.6 Right Triangle Trigonometry © 2010 Pearson Prentice Hall. All rights reserved. 2 Objectives 1. Use the lengths of the sides of a right triangle to find trigonometric ratios. 2. Use trigonometric ratios to find missing parts of right triangles. 3. Use trigonometric ratios to solve applied problems. © 2010 Pearson Prentice Hall. All rights reserved. 3 Ratios in Right Triangles Trigonometry means measurement of triangles. Trigonometric Ratios: Let A represent an acute angle of a right triangle, with right angle, C, shown here. © 2010 Pearson Prentice Hall. All rights reserved. 4 Ratios in Right Triangles For angle A, the trigonometric ratios are defined as follows: © 2010 Pearson Prentice Hall. All rights reserved. 5 Example 1: Becoming Familiar with The Trigonometric Ratios Find the sine, cosine, and tangent of A. Solution: Using the Pythagorean Theorem, find the measure of the hypotenuse c. c² a² b² 5² 12² 25 144 169 c 169 13 Now apply the trigonometric ratios: sinA side opposite angle A 5 hypotenuse 13 side adjacent to angle A 12 cosA hypotenuse 13 side opposite angle A 5 tanA side adjacent to angle A 12 © 2010 Pearson Prentice Hall. All rights reserved. 6 Example 2: Finding a Missing Leg of a Right Triangle Find a in the right triangle Solution: Because we have a known angle, 40°, with a known tangent ratio, and an unknown opposite side, “a,” and a known adjacent side, 150 cm, we can use the tangent ratio. tan 40° = a 150 a = 150 tan 40° ≈ 126 cm © 2010 Pearson Prentice Hall. All rights reserved. 7 Applications of the Trigonometric Ratios • Angle of elevation: Angle formed by a horizontal line and the line of sight to an object that is above the horizontal line. • Angle of depression: Angle formed by a horizontal line and the line of sight to an object that is below the horizontal line. © 2010 Pearson Prentice Hall. All rights reserved. 8 Example 4: Problem Solving using an Angle of Elevation Find the approximate height of this tower. Solution: We have a right triangle with a known angle, 57.2°, an unknown opposite side, and a known adjacent side, 125 ft. Using the tangent ratio: a tan 57.2° = 125 a = 125 tan 57.2° ≈ 194 feet © 2010 Pearson Prentice Hall. All rights reserved. 9 Example 5: Determining the Angle of Elevation A building that is 21 meters tall casts a shadow 25 meters long. Find the angle of elevation of the sun. Solution: We are asked to find mA. © 2010 Pearson Prentice Hall. All rights reserved. 10 Example 5 continued Use the inverse tangent key The display should show approximately 40. Thus the angle of elevation of the sun is approximately 40°. © 2010 Pearson Prentice Hall. All rights reserved. 11