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10-6 Trigonometric Ratios Find the values of the three trigonometric ratios for angle A . 3. 1. SOLUTION: SOLUTION: 4. SOLUTION: 2. SOLUTION: CCSS TOOLS Use a calculator to find the value of each trigonometric ratio to the nearest tenthousandth. 5. sin 37° SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero Keystrokes: 37 Then, sin 37° = 0.6018. 6. cos 23° SOLUTION: Page 1 SOLUTION: : Ratios 37 10-6Keystrokes Trigonometric Then, sin 37° = 0.6018. 6. cos 23° SOLUTION: 10. Keystrokes: 23 Then, cos 23° = 0.9205 SOLUTION: Find the measure of 7. tan 14° . SOLUTION: Keystrokes: 14 Then, tan 14° = 0.2493. Find . 8. cos 82° SOLUTION: Keystrokes: 82 Then, cos 82° = 0.1392. Find . Solve each right triangle. Round each side length to the nearest tenth. 9. SOLUTION: Find the measure of . SOLUTION: Find the measure of Find 11. . . Find . Find . Find . 10. SOLUTION: Find the measure of eSolutions Manual - Powered by Cognero . Page 2 10-6 Trigonometric Ratios The length of the run of the hill is about 11,326.2 ft. Find m X for each right triangle to the nearest degree. 12. SOLUTION: Find the measure of 14. . SOLUTION: –1 Use tan Find on a calculator. Keystrokes: . -1 [TAN ] Find . 15. SOLUTION: 13. SNOWBOARDING A hill used for snowboarding has a vertical drop of 3500 feet. The angle the run makes with the ground is 18°. Estimate the length of r. –1 Use cos on a calculator. Keystrokes: -1 [COS ] SOLUTION: The length of the run of the hill is about 11,326.2 ft. Find m X for each right triangle to the nearest degree. 16. SOLUTION: –1 Use tan on a calculator. Keystrokes: -1 [TAN ] 14. SOLUTION: –1 Use Manual tan on a calculator. eSolutions - Powered by Cognero Keystrokes: -1 [TAN ] Page 3 Keystrokes: Keystrokes: [TAN ] [SIN ] 10-6 Trigonometric Ratios Find the values of the three trigonometric ratios for angle B. 17. SOLUTION: Use sin –1 18. on a calculator. Keystrokes: SOLUTION: Find -1 [SIN ] . Find the values of the three trigonometric ratios for angle B. 18. SOLUTION: Find . 19. SOLUTION: Find . eSolutions Manual - Powered by Cognero Page 4 10-6 Trigonometric Ratios 19. 20. SOLUTION: Find . SOLUTION: Find . CCSS TOOLS Use a calculator to find the value of each trigonometric ratio to the nearest tenthousandth. 21. tan 2° 20. SOLUTION: SOLUTION: Find Keystrokes: 2 Then, tan 2° = 0.0349. . 22. sin 89° SOLUTION: Keystrokes: 89 Then, sin 89° = 0.9998. eSolutions Manual - Powered by Cognero Page 5 23. cos 44° SOLUTION: 28. tan 60° SOLUTION: SOLUTION: Keystrokes: 2 10-6Then, tan 2° = 0.0349. Trigonometric Ratios Keystrokes: 60 Then, tan 60° = 1.7321. 22. sin 89° Solve each right triangle. Round each side length to the nearest tenth. SOLUTION: Keystrokes: 89 Then, sin 89° = 0.9998. 23. cos 44° 29. SOLUTION: SOLUTION: Find the measure of Keystrokes: 44 Then, cos 44° = 0.7193. 24. tan 45° Find SOLUTION: Keystrokes: Then, tan 45° = 1. . . 45 25. sin 73° SOLUTION: Keystrokes: 73 Then, sin 73° = 0.9563 Find . 26. cos 90° SOLUTION: Keystrokes: Then, cos 90° = 0. 90 27. sin 30° 30. SOLUTION: SOLUTION: Find the measure of Keystrokes: 30 Then, sin 30° = 0.5. . 28. tan 60° SOLUTION: Find . Keystrokes: 60 Then, tan 60° = 1.7321. Solve each right triangle. Round each side length to the nearest tenth. Find . 29. SOLUTION: Find the measure of . eSolutions Manual - Powered by Cognero Page 6 31. 10-6 Trigonometric Ratios 31. SOLUTION: Find the measure of . 33. SOLUTION: Find the measure of Find . . Find . Find . Find . 32. SOLUTION: Find the measure of . 34. SOLUTION: Find the measure of . Find . Find . Find . Find eSolutions Manual - Powered by Cognero . Page 7 35. ESCALATORS At a local mall, an escalator is 110 feet long. The angle the escalator makes with the 10-6 Trigonometric Ratios The escalator is about 53 ft high. Find m J for each right triangle to the nearest degree. 34. SOLUTION: Find the measure of 36. . SOLUTION: You know the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. Find . –1 Use a calculator and the [sin ]function to find the measure of the angle. Keystrokes: Find . -1 [SIN ] 10 24 24.62431835 So, m∠J ≈ 25°. 35. ESCALATORS At a local mall, an escalator is 110 feet long. The angle the escalator makes with the ground is 29°. Find the height reached by the escalator. 37. SOLUTION: SOLUTION: You know the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. –1 Use a calculator and the [sin ] function to find the measure of the angle. The escalator is about 53 ft high. Find m J for each right triangle to the nearest degree. Keystrokes: -1 [SIN ] 15 17 61.92751306 So, m∠J ≈ 62°. 36. SOLUTION: eSolutions Powered by Cognero YouManual know -the measure of the side opposite ∠J and the measure of the hypotenuse. Use the sine ratio. Page 8 61.92751306 So, m∠J ≈ 62°. 10-6 Trigonometric Ratios 30.96375653 So, m∠J ≈ 31°. 40. SOLUTION: You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine ratio. 38. SOLUTION: You know the measure of the side opposite ∠J and the measure of the side adjacent to ∠J. Use the tangent ratio. –1 Use a calculator and the [cos ] function to find the measure of the angle. –1 Use a calculator and the [tan ] function to find the measure of the angle. Keystrokes: Keystrokes: -1 [TAN ] 23 -1 [COS ] 5 16 71.79004314 14 So, m∠J ≈ 72°. 58.67130713 So, m∠J ≈ 59°. 41. 39. SOLUTION: You know the measure of the side opposite ∠J and the measure of the side adjacent to ∠J. Use the tangent ratio. SOLUTION: You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine ratio. –1 Use a calculator and the [tan ] function to find the measure of the angle. –1 Use a calculator and the [cos ] function to find the measure of the angle. Keystrokes: -1 [TAN ] 6 10 30.96375653 Keystrokes: -1 [COS ] 11 So, m∠J ≈ 31°. 49.67978493 So, m∠J ≈ 50°. 40. SOLUTION: eSolutions Manual - Powered by Cognero You know the measure of the side adjacent to ∠J and the measure of the hypotenuse. Use the cosine 17 42. MONUMENTS The Lincoln Memorial building measures 204 feet long, 134 feet wide, and 99 feet tall. Chloe is looking at the top of the monument at an angle of 55°. How far away is she standing from the monument? Page 9 SOLUTION: 49.67978493 m∠J ≈ 50°. Ratios 10-6So, Trigonometric 42. MONUMENTS The Lincoln Memorial building measures 204 feet long, 134 feet wide, and 99 feet tall. Chloe is looking at the top of the monument at an angle of 55°. How far away is she standing from the monument? SOLUTION: The horizontal distance to the city is about 35,577 ft. 44. FORESTS A forest ranger estimates the height of a tree is about 175 feet. If the forest ranger is standing 100 feet from the base of the tree, what is the measure of the angle formed between the range and the top of the tree? SOLUTION: The angle formed between the ground and the top of the tree is about 60°. Chloe is standing about 69 ft away from the monument. 43. AIRPLANES Ella looks down at a city from an airplane window. The airplane is 5000 feet in the air, and she looks down at an angle of 8°. Determine the horizontal distance to the city. Suppose ABC . A is an acute angle of right triangle 45. Find sin A and tan A if cos . SOLUTION: SOLUTION: The horizontal distance to the city is about 35,577 ft. 44. FORESTS A forest ranger estimates the height of a tree is about 175 feet. If the forest ranger is standing 100 feet from the base of the tree, what is the measure of the angle formed between the range and the top of the tree? eSolutions Manual - Powered by Cognero SOLUTION: 46. Find tan A and cos A if sin SOLUTION: . Page 10 10-6 Trigonometric Ratios 46. Find tan A and cos A if sin . . SOLUTION: SOLUTION: 47. Find cos A and tan A if sin 48. Find sin A and cos A if tan . SOLUTION: 49. SUBMARINES A submarine descends into the ocean at an angle of 10° below the water line and travels 3 miles diagonally. How far beneath the surface of the water has the submarine reached? SOLUTION: The submarine went about 0.5 mi beneath the surface of the water. 48. Find sin A and cos A if tan . 50. MULTIPLE REPRESENTATIONS In this problem, you will explore a relationship between the sine and cosine functions. SOLUTION: a. TABULAR Copy and complete the table using the triangles shown above. eSolutions Manual - Powered by Cognero Page 11 b. VERBAL Make a conjecture about the sum of 10-6 Trigonometric Ratios a. TABULAR Copy and complete the table using the triangles shown above. b. The sum of the squares of the sine and cosine of an acute angle in a right triangle is equal to 1. 51. CHALLENGE Find a and c in the triangle shown. b. VERBAL Make a conjecture about the sum of the squares of the sine and cosine of an acute angle of an acute angle in a right triangle. SOLUTION: a. SOLUTION: Use the sum of the angles of a triangle to determine the value of a. Use the value of a to determine the value of the angles of the triangle. The angles are 90°, 6(5) – 3 or 27°, and 12(5) + 3 or 63°. Use a trigonometric ratio to find the value of c. Therefore, a = 5 and c ≈ 7.3. Then , 52. REASONING Use the definitions of the sine and cosine ratios to define the tangent ratio. SOLUTION: Sin is defined as and Cos is defined as Tan can be defined as . because: b. The sum of the squares of the sine and cosine of an acute angle in a right triangle is equal to 1. 51. CHALLENGE Find a and c in the triangle shown. eSolutions Manual - Powered by Cognero Page 12 Therefore, a = 5 and c ≈ 7.3. 10-6 Trigonometric Ratios 52. REASONING Use the definitions of the sine and cosine ratios to define the tangent ratio. SOLUTION: Sin is defined as and Cos is defined as Tan can be defined as . 54. CCSS ARGUMENTS The sine and cosine of an acute angle in a right triangle are equal. What can you conclude about the triangle? SOLUTION: Given: ΔABC with sides a, b, and c as shown; sin A = cos A because: 53. WRITING IN MATH How can triangles be used to solve problems? SOLUTION: Many real world problems involve trying to determine the correct height or length of a given structure. When lengths and angles are known, right triangles can be drawn, and trigonometric ratios can be used to determine missing sides and angles. Similarly, other situations may require triangles and the Pythagorean theorem to determine unknown lengths. If a = b, then . A triangle that has two congruent sides is called an isosceles triangle. Therefore, this triangle is an isosceles right triangle. The legs of the right triangle are equal to each other. 55. WRITING IN MATH Explain how to use trigonometric ratios to find the missing length of a side of a right triangle given the measure of one acute angle and the length of one side. SOLUTION: Use the acute angle given and the measure of the known side to set up one of the trigonometric ratios. The sine ratio uses the opposite side and hypotenuse of the triangle. The cosine ratio uses the adjacent side and hypotenuse of the triangle. The tangent ratio uses the opposite and adjacent sides of the triangle. Choose the ratio that can be used to solve for the unknown measure. Given the following triangle find the missing sides a and c. 54. CCSS ARGUMENTS The sine and cosine of an acute angle in a right triangle are equal. What can you conclude about the triangle? eSolutions Manual - Powered by Cognero SOLUTION: Given: ΔABC with sides a, b, and c as shown; sin A = cos A Page 13 Since you know the measure of ∠A, set up the uses the opposite and adjacent sides of the triangle. Choose the ratio that can be used to solve for the unknown measure. 10-6 Trigonometric Ratios Given the following triangle find the missing sides a and c. 56. Which graph below represents the solution set for −2 ≤ x ≤ 4? A B C Since you know the measure of ∠A, set up the trigonometric ratios for the acute angle of 42°. Let a be the measure of the side opposite ∠A, 15 is the measure of the side adjacent ∠A, and c is the measure of the hypotenuse. So, if you are trying to find the measure of a, use the tangent ratio. If you are trying to find the measure of c, use the cosine ratio. D SOLUTION: The inequality uses less than or equal to signs, so the points on the graph must be solid. So, choices B and D are incorrect. In the inequality, x is found between the two values, so choice C in incorrect. The correct choice is A. 57. PROBABILITY Suppose one chip is chosen from a bin with the chips shown. To the nearest tenth, what is the probability that a green chip is chosen? F 0.2 G 0.5 H 0.6 J 0.8 56. Which graph below represents the solution set for −2 ≤ x ≤ 4? A SOLUTION: B C The correct choice is F. 58. In the graph, for what value(s) of x is y = 0? D SOLUTION: eSolutions Manual - Powered by Cognero The inequality uses less than or equal to signs, so the points on the graph must be solid. So, choices B and Page 14 10-6 Trigonometric Ratios The correct choice is F. a. Let h represent the height reached by the ladder. Use the Pythagorean Theorem to represent the value of h in terms of the other two sides. 58. In the graph, for what value(s) of x is y = 0? If the bottom of the ladder is moved closer to the base of the house, the distance the bottom of the 2 A 0 B −1 C 1 D 1 and −1 SOLUTION: The graph crosses the x-axis twice, so there are two values of x for which y = 0. Choices A, B, and C only offer one x value. The correct choice is D. 59. EXTENDED RESPONSE A 16-foot ladder is placed against the side of a house so that the bottom of the ladder is 8 feet from the base of the house. a. If the bottom of the ladder is moved closer to the base of the house, does the height reached by the ladder increase or decrease? b. What conclusion can you make about the distance between the bottom of the ladder and the base of the house and the height reached by the ladder? c. How high does the ladder reach if the ladder is 3 feet from the base of the house? ladder is from the wall will decrease. When 16 is 2 subtracted by a number smaller than 8 , the 2 2 difference is greater than 16 - 8 . Since you are finding the square root of a larger number, h will be greater. Therefore, as the bottom of the ladder is moved closer to the base of the house, the height reached by the ladder will increase. b. Sample answer: Let h represent the height reached by the ladder and d represent the distance between the bottom of the ladder and the base of the house. The house is built perpendicular to the ground, so the ladder will form a right triangle when it is placed against the side of the house. Use the Pythagorean Theorem to relate the sides of the triangle. 2 Therefore, the sum of their squares is 16 or 256. c. SOLUTION: If the ladder is 3 ft from the base of the house, then it reaches a height of about 15.7 ft. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 60. a = 16, b = 63, c = ? SOLUTION: a. Let h represent the height reached by the ladder. Use the Pythagorean Theorem to represent the value of h in terms of the other two sides. eSolutions Manual - Powered by Cognero Page 15 The length of the hypotenuse is 65 units. If the bottom of the ladder is moved closer to the The length of one of the legs is units. the ladder is 3 ftRatios from the base of the house, then it 10-6IfTrigonometric reaches a height of about 15.7 ft. If c is the measure of the hypotenuse of a right triangle, find each missing measure. If necessary, round to the nearest hundredth. 60. a = 16, b = 63, c = ? or about 10.72 63. a = 6, b = 3, c = ? SOLUTION: SOLUTION: The length of the hypotenuse is units. The length of the hypotenuse is 65 units. 64. 61. b = 3, ,c=? or about 6.71 , c = 12, a = ? SOLUTION: SOLUTION: The length of the hypotenuse is 11 units. The length of one of the legs is units. or about 8.19 62. c = 14, a = 9, b = ? 65. a = 4, SOLUTION: The length of one of the legs is units. 63. a = 6, b = 3, c = ? SOLUTION: ,c=? SOLUTION: or about 10.72 The length of the hypotenuse is units. or about 5.20 66. AVIATION The relationship between a plane’s length L in feet and the pounds P its wings can lift is eSolutions Manual - Powered by Cognero described by , where k is the constant of proportionality. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine k Page 16 for this plane to the nearest hundredth. SOLUTION: length of the hypotenuse is 10-6The Trigonometric Ratios units. or about 5.20 66. AVIATION The relationship between a plane’s length L in feet and the pounds P its wings can lift is 69. described by , where k is the constant of proportionality. A Boeing 747 is 232 feet long and has a takeoff weight of 870,000 pounds. Determine k for this plane to the nearest hundredth. SOLUTION: SOLUTION: 70. The constant of proportionality is about 0.06. SOLUTION: 67. FINANCIAL LITERACY A salesperson is paid $32,000 a year plus 5% of the amount in sales made. What is the amount of sales needed to have an annual income greater than $45,000? SOLUTION: Let x represent the amount of sales made. 71. The amount of sales must be more than $260,000. Solve each proportion. SOLUTION: 68. SOLUTION: 69. SOLUTION: eSolutions Manual - Powered by Cognero Page 17