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Trigonometric Ratios: Angles of Elevation and Depression

Learn how to use angles of elevation and depression to find distances between objects using trigonometric ratios in applied problems.

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Trigonometric Ratios: Angles of Elevation and Depression

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 8–4) CCSS Then/Now New Vocabulary Example 1: Angle of Elevation Example 2: Angle of Depression Example 3: Use Two Angles of Elevation or Depression Lesson Menu

  3. Use a calculator to find tan 54°. A. 0.6738 B. 0.7432 C. 0.8090 D. 1.3764 5-Minute Check 1

  4. Find mB to the nearest tenth of a degree if cos B = 0.8926 and B is an acute angle. A. 63.2 B. 26.8 C. 18.4 D. 0.999 5-Minute Check 2

  5. Find x. Round to the nearest tenth. A. 14.1 B. 17.4 C. 19.4 D. 21.3 5-Minute Check 3

  6. Find x. Round to the nearest tenth. A. 9.5 B. 15.9 C. 23.7 D. 30.8 5-Minute Check 4

  7. Find x. Round to the nearest tenth. A. 34.7 B. 43.8 C. 46.2 D. 52.5 5-Minute Check 5

  8. A. B. C. D. Use a special right triangle to express sin 45° as a fraction. 5-Minute Check 6

  9. Content Standards G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Mathematical Practices 4 Model with mathematics. 1 Make sense of problems and persevere in solving them. CCSS

  10. You used similar triangles to measure distances indirectly. • Solve problems involving angles of elevation and depression. • Use angles of elevation and depression to find the distance between two objects. Then/Now

  11. angle of elevation • angle of depression Vocabulary

  12. Angle of Elevation CIRCUS ACTS At the circus, a person in the audience at ground level watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°? Make a drawing. Example 1

  13. Divide both sides by tan Angle of Elevation Since QR is 25 feet and RS is 5 feet 6 inches or 5.5 feet, QS is 30.5 feet. Let x represent PQ. Multiply both sides by x. Simplify. Example 1

  14. Angle of Elevation Answer: The audience member is about 60 feet from the base of the platform. Example 1

  15. DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree? A. 37° B. 35° C. 40° D. 50° Example 1

  16. Since are parallel, mBAC = mACD by the Alternate Interior Angles Theorem. Angle of Depression DISTANCE Maria is at the top of a cliff and sees a seal in the water. If the cliff is 40 feet above the water and the angle of depression is 52°, what is the horizontal distance from the seal to the cliff, to the nearest foot? Make a sketch of the situation. Example 2

  17. Angle of Depression Let x represent the horizontal distance from the seal to the cliff, DC. C = 52°; AD = 40, and DC = x Multiply each side by x. Example 2

  18. Angle of Depression Divide each side by tan 52°. Answer: The seal is about 31 feet from the cliff. Example 2

  19. Luisa is in a hot air balloon 30 feet above the ground. She sees the landing spot at an angle of depression of 34. What is the horizontal distance between the hot air balloon and the landing spot to the nearest foot? A. 19 ft B. 20 ft C. 44 ft D. 58 ft Example 2

  20. Use Two Angles of Elevation or Depression DISTANCE Vernon is on the top deck of a cruise ship and observes two dolphins following each other directly away from the ship in a straight line. Vernon’s position is 154 meters above sea level, and the angles of depression to the two dolphins are 35° and 36°. Find the distance between the two dolphins to the nearest meter. Example 3

  21. Plan Because are horizontal lines, they are parallel. Thus, and because they are alternate interior angles. This means that Use Two Angles of Elevation or Depression Understand ΔMLK andΔMLJare right triangles. The distance between the dolphins isJK or JL – KL. Use the right triangles to find these two lengths. Example 3

  22. Divide each side by tan Use Two Angles of Elevation or Depression Solve Multiply each side by JL. Use a calculator. Example 3

  23. Divide each side by tan Use Two Angles of Elevation or Depression Multiply each side by KL. Use a calculator. Answer: The distance between the dolphins is JK – KL. JL – KL≈ 219.93 – 211.96, or about 8 meters. Example 3

  24. Madison looks out her second-floor window, which is 15 feet above the ground. She observes two parked cars. One car is parked along the curb directly in front of her window and the other car is parked directly across the street from the first car. The angles of depression of Madison’s line of sight to the cars are 17° and 31°. Find the distance between the two cars to the nearest foot. A. 14 ft B. 24 ft C. 37 ft D. 49 ft Example 3

  25. End of the Lesson

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