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Page 1 of 6 10.4 Goal Find the tangent of an acute angle. Key Words • trigonometric ratio • leg opposite an angle Tangent Ratio How can you find the height of the tree in the photograph at the right? It is too tall to be measured directly in any simple manner. You can determine the height using a trigonometric ratio. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. • leg adjacent to an angle • tangent leg opposite angle leg adjacent For any acute angle of a right triangle, there is a leg opposite the angle and a leg adjacent to the angle. The ratio of these legs is the tangent of the angle. TANGENT RATIO For any acute angle A of a right triangle: B hypotenuse Student Help READING TIP leg opposite aA a tan A b leg adjacent to aA The tangent of angle A is written as “tan A.” A EXAMPLE 1 leg a opposite aA c b leg adjacent to aA C Find Tangent Ratio Find tan S and tan R as fractions in simplified form and as decimals rounded to four decimal places. R Solution leg opposite aS leg adjacent to aS 43 4 leg opposite aR leg adjacent to aR 4 43 S 8 43 4 T tan S 3 ≈ 1.7321 1 3 tan R ≈ 0.5774 10.4 Tangent Ratio 557 Page 2 of 6 Tangent Function You can use the TAN function on a calculator to approximate the tangent of an angle. You can also use the table of trigonometric ratios on page 705. Student Help EXAMPLE KEYSTROKE HELP When calculating the tangent of an angle, be sure your calculator is in DEGREE mode. Use a Calculator for Tangent 2 Approximate tan 74 to four decimal places. Solution Calculator keystrokes 74 or 74 Display Rounded value 3.487414444 3.4874 Find Tangent Ratio Find tan S and tan R as fractions in simplified form and as decimals. Round to four decimal places if necessary. 1. 10 S 8 R 2. R 6 10 T T 26 24 S Use a calculator to approximate the value to four decimal places. 3. tan 35 EXAMPLE 4. tan 85 5. tan 10 Find Leg Length 3 Use a tangent ratio to find the value of x. Round your answer to the nearest tenth. 3 22 x Solution opposite leg adjacent leg Write the tangent ratio. 3 x Substitute. tan 22 tan 22 x p tan 22 3 3 tan 22 x Chapter 10 Divide each side by tan 22. x ≈ Use a calculator or table to approximate tan 22. x ≈ 7.4 Simplify. 3 0.4040 558 Multiply each side by x. Right Triangles and Trigonometry Page 3 of 6 In Example 3, the unknown, x, was in the denominator of the ratio. If you prefer to use a ratio in which the unknown is in the numerator, use the tangent ratio for the other acute angle in the triangle. Student Help STUDY TIP Given the measure of one acute angle of a right triangle, use the Corollary to the Triangle Sum Theorem to find the measure of the other angle. EXAMPLE Find Leg Length 4 Use two different tangent ratios to find the value of x to the nearest tenth. 4 35 x Solution First, find the measure of the other acute angle: 90 35 55. Method 1 opposite leg tan 35 adjacent leg Method 2 opposite leg tan 55 adjacent leg x 4 tan 35 4 x tan 55 x p tan 35 4 4 tan 55 x 4 tan 35 4(1.4281) ≈ x x 4 0.7002 x ≈ x ≈ 5.7 x ≈ 5.7 ANSWER The two methods yield the same answer: x ≈ 5.7. Careers EXAMPLE Estimate Height 5 You stand 45 feet from the base of a tree and look up at the top of the tree as shown in the diagram. Use a tangent ratio to estimate the height of the tree to the nearest foot. h Solution FORESTERS Foresters manage and protect forests. To determine the height of a tree, a forester may use trigonometry. tan 59 opposite leg adjacent leg Write ratio. h 45 Substitute. tan 59 45 tan 59 h Multiply each side by 45. 45(1.6643) ≈ h Use a calculator or table to approximate tan 59. 74.9 ≈ h Career Links CLASSZONE.COM ANSWER Simplify. The tree is about 75 feet tall. 10.4 Tangent Ratio 559 Page 4 of 6 Find Side Length Write two equations you can use to find the value of x. 6. 7. 8. x 44 4 5 37 59 x x 8 Find the value of x. Round your answer to the nearest tenth. 9. 10. 11. 18 20 60 7 x 34 x 55 x 10.4 Exercises Guided Practice Vocabulary Check 1. Name the acute angles in T DEF. E 2. Identify the leg opposite aD and the D leg adjacent to aD. Skill Check F Find tan A as a fraction in simplified form and as a decimal. Round to four decimal places if necessary. 3. A 9 C 4. 15 12 5. A A 52 5 83 C 8 16 B B B 5 C Use a calculator to approximate the value to four decimal places. 6. tan 25 7. tan 62 8. tan 80 9. tan 43 Practice and Applications Extra Practice Finding Tangent Ratios Find tan A. Write your answer as a fraction. See p. 693. 10. B 8 C 560 Chapter 10 11. A 17 15 Right Triangles and Trigonometry 12 A B 63 6 C 12. C 2 B 3 13 A Page 5 of 6 Homework Help Example 1: Exs. 10–15 Example 2: Exs. 16–23 Example 3: Exs. 24–29, 32–37 Example 4: Exs. 24–29, 32–37 Example 5: Exs. 31, 38 Finding Tangent Ratios Find tan P and tan R as fractions in simplified form and as decimals rounded to four decimal places. 13. P 24 14. 25 37 R 12 35 P P 15. P 15 20 P 25 R P 7 R Using a Calculator Use a calculator to approximate the value to four decimal places. 16. tan 28 17. tan 54 18. tan 5 19. tan 89 20. tan 67 21. tan 40 22. tan 12 23. tan 83 Using Tangent Ratios Write two equations you can use to find the value of x. Then find the value of x to the nearest tenth. 24. 25. 35 x 39 9 x 26. 56 x 70 33 Finding Leg Lengths Find the value of x to the nearest tenth. 27. x 28. 29. 20 29 52 12 53 41 x x 30. Error Analysis To find the length B &* in the diagram at the right, of BC Water Slides 18 BC a student wrote tan 55 . Explain the student’s error. A 18 30 55 C 31. Water Slide A water slide makes an angle of about 13 with the ground. The slide extends horizontally about 58.2 meters as shown below. Find the height h of the slide to the nearest tenth of a meter. Not drawn to scale WATER SLIDES Riders on some of the newer water slides may travel at speeds greater than 50 mi/h. Application Links 58.2 meters CLASSZONE.COM 10.4 Tangent Ratio 561 Page 6 of 6 Finding Side Lengths Find the value of x to the nearest tenth. IStudent Help ICLASSZONE.COM 32. HOMEWORK HELP Extra help with problem solving in Exs. 32–37 is at classzone.com 33. x 49 34. x x 70 34 10 14 3 35. 35 36. x 37. 20 x 45 x 40 35 50 10 38. Surveying To find the distance d from a house on shore to a house on an island, a surveyor measures from the house on shore to point B, as shown in the diagram. An instrument called a transit is used to find the measure of aB. Find the distance d to the nearest tenth of a meter. Standardized Test Practice 39. Multiple Choice Which expression can be used to find the value of x in the triangle shown? A x 10 tan 38 10 C x tan 38 tan 38 B x 10 10 10 D x tan 52 38 x 40. Multiple Choice What is the approximate value of y in the triangle shown? F H Mixed Review G J 7.2 9.3 8.4 50 6 10.1 y Volume of Solids Find the volume of the solid. If necessary, round your answer to the nearest whole number. (Lessons 9.4, 9.5) 41. 42. 6m 43. 14 in. 8 ft 9m 4 ft Algebra Skills 562 Chapter 10 10 in. 5 ft Solving Equations Solve the equation. (Skills Review, p. 673) 44. 8x 10 3x 45. 4(x 3) 32 46. 3x 7 x 11 47. 6x 5 3x 4 48. 2 x 4x 22 49. 5x 18 2x 21 Right Triangles and Trigonometry