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Page 1 of 6 10.5 Goal Find the sine and cosine of an acute angle. Sine and Cosine Ratios Two other trigonometric ratios are sine and cosine . These are abbreviated as sin and cos. Unlike the tangent ratio, these ratios involve the hypotenuse of a right triangle. Key Words SINE AND COSINE RATIOS • sine • cosine For any acute angle A of a right triangle: B leg opposite aA a sin A hypotenuse c leg adjacent to aA hypotenuse b c cos A 1 EXAMPLE hypotenuse c leg a opposite aA A C b leg adjacent to aA Find Sine and Cosine Ratios Find sin A and cos A. B 5 A Solution leg opposite a A sin A hypotenuse 3 C 4 Write ratio for sine. 3 5 Substitute. leg adjacent to a A hypotenuse Write ratio for cosine. 4 5 Substitute. cos A Find Sine and Cosine Ratios Find sin A and cos A. 1. B 17 15 2. B 24 25 C 3. B 7 A 8 C A 8 10 6 A C 10.5 Sine and Cosine Ratios 563 Page 2 of 6 2 EXAMPLE Find Sine and Cosine Ratios Find sin A and cos A. Write your answers as fractions and as decimals rounded to four decimal places. Solution B 13 leg opposite a A 5 13 sin A ≈ 0.3846 hypotenuse leg adjacent to a A hypotenuse A 5 12 C 12 13 cos A ≈ 0.9231 Find Sine and Cosine Ratios Find sin A and cos A. Write your answers as fractions and as decimals rounded to four decimal places. 4. C 40 9 A B 5. 6. C 39 2 5 8 C A B 2 41 A 2 B Sine and Cosine Functions You can use the SIN and COS functions on a calculator to approximate the sine and cosine of an angle. You can also use the table of trigonometric ratios on page 705. EXAMPLE 3 Use a Calculator for Sine and Cosine Use a calculator to approximate sin 74 and cos 74. Round your answers to four decimal places. Solution Calculator keystrokes Display Rounded value 74 or 74 0.961261696 0.9613 74 or 74 0.275637356 0.2756 Use a Calculator for Sine and Cosine Use a calculator to approximate the value to four decimal places. 564 Chapter 10 7. sin 43 8. cos 43 9. sin 15 10. cos 15 11. cos 72 12. sin 72 13. cos 90 14. sin 90 Right Triangles and Trigonometry Page 3 of 6 Find Leg Lengths 4 EXAMPLE Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. B 10 a 32 A Student Help VISUAL STRATEGY leg opposite a A A 32 8.5 C 58 5.3 cos A a 10 b 10 cos 32 sin 32 C You can label side lengths and angle measures in different colors. See p. 536. leg adjacent to a A hypotenuse sin A hypotenuse B 10 b Solution 10(sin 32) a 10(cos 32) b 10(0.5299) ≈ a 10(0.8480) ≈ b 5.3 ≈ a 8.5 ≈ b ANSWER In the triangle, BC is about 5.3 and AC is about 8.5. Find Leg Lengths Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. 15. 16. 7 a b 65 17. 12 43 a a 34 b 5 b SUMMARY TRIGONOMETRIC RATIOS For any acute angle A of a right triangle: B Tangent of aA c leg opposite aA a tan A leg adjacent to aA b A a b C B Sine of aA c leg opposite aA a sin A hypotenuse c A b a C B Cosine of aA c leg adjacent to aA b cos A hypotenuse c A 10.5 b a C Sine and Cosine Ratios 565 Page 4 of 6 10.5 Exercises Guided Practice Vocabulary Check Use the diagram shown at the right to match the trigonometric ratios. EF DE 1. cos D A. 2. sin D B. 3. tan D C. F DE DF D EF DF 4. Error Analysis A student says E B that sin D > sin A because the side lengths of T DEF are greater than the side lengths of T ABC. Explain why the student is incorrect. Skill Check E A 37° C 37° D F In Exercises 5–10, use the diagram shown below to find the trigonometric ratio. 5. sin A 6. cos A 7. tan A 8. sin B 9. cos B 10. tan B A 5 3 C 4 B Practice and Applications Extra Practice Finding Sine and Cosine Ratios Find sin A and cos A. Write your answers as fractions in simplest form. See p. 694. 11. B 11 C 12. C 61 60 53 B 13. A 5 10 B A Exs. 11–13 Exs. 14–16 Exs. 17–24 Exs. 25–30 14. P 12 35 P 15. R 11 37 Chapter 10 62 R P 566 36 15 C Finding Sine and Cosine Ratios Find sin P and cos P. Write your answers as fractions in simplest form and as decimals rounded to four decimal places. Homework Help Example 1: Example 2: Example 3: Example 4: A 39 Right Triangles and Trigonometry 7 P 16. 9 P 62 R 3 P Page 5 of 6 Calculator Use a calculator to approximate the value to four decimal places. IStudent Help ICLASSZONE.COM HOMEWORK HELP Extra help with problem solving in Exs. 25–30 is at classzone.com 17. sin 40 18. cos 23 19. sin 80 20. cos 5 21. sin 59 22. cos 61 23. sin 90 24. cos 77 Finding Leg Lengths Find the lengths of the legs of the triangle. Round your answers to the nearest tenth. 25. 26. 8 x 16 24 x Skateboarding x 11 29 y y 31. 54 y 46 28. x 27. 14 x 29. 30. 26 y 15 37 68 y x y Visualize It! A ladder that is 15 feet long is leaning against a wall. The ladder makes an angle of 70 with the ground. Make a sketch. Then determine how high up the wall the ladder reaches. Round your answer to the nearest foot. 32. Skateboard Ramp You are constructing a skateboarding ramp like the one shown below. Your ramp will be 8 feet long and the ramp angle will be about 22. Find the lengths of the legs of the triangles that support the ramp. Round your answers to the nearest inch. SKATEBOARD RAMPS built with curves, called half pipes, allow freestyle skateboard riders to perform acrobatic maneuvers. Technology In Exercises 33–35, use geometry software. &*. 1 Draw AB ● &* through B. 2 Construct a perpendicular to AB ● 3 Add point C on the perpendicular. ● &*. 4 Draw AC ● 33. Find maA, sin A, and cos A. C A B 34. Calculate (sin A)2 (cos A)2. 35. Drag point C. What do you notice? 10.5 Sine and Cosine Ratios 567 Page 6 of 6 36. Challenge Let A be any acute angle of a right triangle. Use the definitions of sin A, cos A, and tan A to prove the following result. sin A tan A cos A You be the Judge 37. One student uses R r &*. the ratio sin 42 to find the length of ST 34 r Another student uses the ratio cos 48 . 34 Assuming the students make no errors in calculation, who will get the correct answer? Standardized Test Practice 42 34 S r T 38. Multiple Choice Use the diagram below. Which expression could be used to find CD? A C 8(cos 25) 8 sin 25 D B D 8(sin 25) 8 cos 25 8 25 C E 39. Multiple Choice Which statement cannot be true? F H Mixed Review s sin A 0.55 sin A 1.2 G J B sin A 0.61 sin A 0.4869 A C Not drawn to scale Finding Leg Lengths Find the value of each variable. Write your answer in radical form. (Lesson 10.3) 40. 41. 13 60 42. 26 30 x x 30 y y 30 16 60 x 60 7 Using a Calculator Use a calculator to approximate the value to four decimal places. (Lesson 10.4) Algebra Skills 43. tan 32 44. tan 88 45. tan 56 46. tan 24 47. tan 17 48. tan 49 Ordering Numbers Write the numbers in order from least to greatest. (Skills Review, p. 662) 49. 0.8, 1.8, 8, 0.08, 18, 0, 1.8 50. 2641, 2146, 2614, 2416, 2164, 2461 51. 0.56, 0.47, 0.61, 0.5, 0.6 568 Chapter 10 Right Triangles and Trigonometry