1. Slide 5.2- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Right Triangle Trigonometry Learn the definitions of the trigonometric functions of acute angles. Learn to evaluate trigonometric functions of acute angles. Learn the values of the trigonometric functions for the special angles 30º, 45º, and 60º. Learn to use right triangle trigonometry in applications. SECTION 5.2 1 2 3 4

3. Slide 5.2- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC RATIOS AND FUNCTIONS a = length of the side opposite  b = length of the side adjacent to  c = length of the hypotenuse

4. Slide 5.2- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RIGHT TRIANGLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 

5. Slide 5.2- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding the Values of Trigonometric Functions Find the exact values for the six trigonometric functions of the angle  in the figure. Solution

6. Slide 5.2- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding the Values of Trigonometric Functions Solution continued

7. Slide 5.2- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC FUNCTION VALUES OF SOME COMMON ANGLES

8. Slide 5.2- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC FUNCTION VALUES OF SOME COMMON ANGLES  deg  radians

9. Slide 5.2- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley COMPLEMENTARY ANGLES The value of any trigonometric function of an acute angle  is equal to the cofunction of the complement of . This is true whether  is measured in degrees or in radians. If  is measured in radians, replace 90º with  in degrees

10. Slide 5.2- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Finding Trigonometric Function Values of a Complementary Angle Solution

11. Slide 5.2- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Measuring the Height of Mount Kilimanjaro A surveyor wants to measure the height of Mount Kilimanjaro by using the known height of a nearby mountain. The nearby location is at an altitude of 8720 feet, the distance between that location and Mount Kilimanjaro’s peak is 4.9941 miles, and the angle of elevation from the lower location is 23.75º. See the figure on the next slide. Use this information to find the approximate height of Mount Kilimanjaro.

12. Slide 5.2- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Measuring the Height of Mount Kilimanjaro

13. Slide 5.2- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Measuring the Height of Mount Kilimanjaro Solution The sum of the side length h and the location height of 8720 feet gives the approximate height of Mount Kilimanjaro. Let h be measured in miles. Use the definition of sin , for  = 23.75º.

14. Slide 5.2- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Measuring the Height of Mount Kilimanjaro Solution continued 1 mile = 5280 feet Thus, the height of Mount Kilimanjaro

15. Slide 5.2- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Width of a River To find the width of a river, a surveyor sights straight across the river from a point A on her side to a point B on the opposite side. See the figure on the next slide. She then walks 200 feet upstream to a point C. The angle  that the line of sight from point C to point B makes with the river bank is 58º. How wide is the river?

16. Slide 5.2- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Width of a River

17. Slide 5.2- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding the Width of a River The river is about 320 feet wide at the point A. A, B, and C are the vertices of a right triangle with acute angle 58º. w is the width of the river. Solution