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Chapter 13 – Trigonometric Ratios 13.1 Answer Key Pythagorean Theorem and its Converse Answers 1. 194 2. 63 3. 5 3 4. c = 10 5. 4 10 6. 6 5 7. Yes 8. No 9. No 10. Yes 11. No 12. No 13. 1 1 (b + a )( a + b ) → ( a 2 + 2ab + b2 ) 2 2 14. 1 ⎛1 ⎞ 1 2 ⎜ ab ⎟ + c 2 → ( 2ab + c 2 ) 2 ⎝2 ⎠ 2 15. Students must provide proof. CK-‐12 Algebra II with Trigonometry Concepts 1 Chapter 13 – Trigonometric Ratios 13.2 Answer Key Sine, Cosine and Tangent Answers 1. 0.5736 2. 3.0777 3. 0.6691 4. 1 5. 0.5 6. 0.0349 7. 3 4 3 4 3 4 sin N = , cos N = , tan N = ; sin M = , cos M = , tan M = 5 5 4 5 5 3 8. x ≈ 5.14, y ≈ 6.13 9. x ≈ 11.03, y ≈ 4.66 10. x ≈ 8.66, y ≈ 10 11. b ≈ 17.60, c ≈ 12.87 12. c ≈ 23.18, a ≈ 6.21 13. a ≈ 15.01, b ≈ 16.56 14. 17.3 ft 15. 12.7 km CK-‐12 Algebra II with Trigonometry Concepts 2 Chapter 13 – Trigonometric Ratios 13.3 Answer Key Inverse Trig Functions and Solving Right Triangles Answers 1. 44° 2. 60° 3. 81° 4. 13° 5. 61° 6. 20° 7. x ≈ 45°, y ≈ 45° 8. x ≈ 71°, y ≈ 19° 9. x ≈ 27°, y ≈ 63° 10. x ≈ 45°, y ≈ 45° 11. x ≈ 50°, y ≈ 40° 12. x ≈ 33°, y ≈ 57° 13. m∠B ≈ 41°, m∠A ≈ 39°, b ≈ 39.7 14. m∠B ≈ 56°, m∠A ≈ 24°, c ≈ 10.8 15. m∠B ≈ 40°, m∠A ≈ 50°, a ≈ 10.7 CK-‐12 Algebra II with Trigonometry Concepts 3 Chapter 13 – Trigonometric Ratios 13.4 Answer Key Application Problems Answers 1. 11.3 in 2. 477 m 3. 35 m 4. 97 ft 5. 39° 6. 88 ft 7. 31° 8. 97 ft 9. 29 ft 10. 13 miles 11. The hypotenuse is always the longest side. Therefore, the ratios, CK-‐12 Algebra II with Trigonometry Concepts O A < 1 and < 1. H H 4 Chapter 13 – Trigonometric Ratios 13.5 Answer Key Introduction to Angles of Rotations, Coterminal Angles and Reference Angles Answers 1. −458° , 262° 2. 115° , −245° 3. −570° , 150° 4. −313° , 407° 5. −302° , 58° 6. −6°, 714° 7. 353°, − 367° 8. QII, 78° 9. QIV, 40° 10. QIII, 47° 11. QIII, 80° 12. QIV, 56° 13. QIII, 71° 14. QIV, 12° 15. All the angles between 0° and 90° are acute angles between the terminal side of the angle and the x-axis. CK-‐12 Algebra II with Trigonometry Concepts 5 Chapter 13 – Trigonometric Ratios 13.6 Answer Key Introduction to the Unit Circle and Radian Measure Answers 1. 3π 4 2. 4π 3 3. − 4. 5π 2 5. − 6. 420° 7. −390° 8. 810° 9. −135° 10. 150° 11. coterminal angles: π 2π 4π ; reference angle: QII ,− 3 3 3 12. coterminal angles: π 3π 5π ; reference angle: QII ,− 4 4 4 13. coterminal angles: π 11π 13π ; reference angle: QIV ,− 6 6 6 14. coterminal angles: π 10π 2π ; reference angle: QIII ,− 3 3 3 15. coterminal angles: − 11π 6 7π 4 π 5π 7π ; reference angle: QIII , 6 6 6 CK-‐12 Algebra II with Trigonometry Concepts 6 Chapter 13 – Trigonometric Ratios 13.7 Answer Key Trigonometric Ratios on the Unit Circle Answers 2 2 1. 2. 0 3. − 3 4. 1 2 5. − 6. 0 2 2 7. 8. 1 2 − 2 2 3 3 9. 2 2 10. − 11. 0 12. 1 13. − 14. Undefined 15. 1 2 3 2 CK-‐12 Algebra II with Trigonometry Concepts 7 Chapter 13 – Trigonometric Ratios 13.8 Answer Key Reciprocal Trigonometric Functions Answers 1. 1.0038 2. -0.1405 3. -1.2361 4. -0.4663 5. -1.1099 6. -1.5080 7. -1.9626 8. -1.7013 9. − 10. 1 11. 1 12. 2 13. -2 14. Undefined 15. − 16. − 2 2 3 3 3 3 CK-‐12 Algebra II with Trigonometry Concepts 8 Chapter 13 – Trigonometric Ratios 13.9 Answer Key Inverse Trigonometric Functions Answers 1. 102.6° , 257.4° 2. 84.7° , 275.3° 3. 92.8° , 272.8° 4. 61.5° , 118.5° 5. 188.3° , 351.7° 6. 50.3° , 230.3° 7. 3.80, 5.62 8. 1.43, 4.85 9. 2.80, 5.94 10. 1.68, 4.82 11. 0.78, 3.92 12. 0.08, 3.06 13. 0, 14. 3π 5π , 4 4 15. 3π 7π , 4 4 16. π 11π , 6 6 17. π 5π , 6 6 18. 0, π 19. 2π 4π , 3 3 20. π 3π , 4 4 21. π 7π , 6 6 π CK-‐12 Algebra II with Trigonometry Concepts 9 Chapter 13 – Trigonometric Ratios 13.10 Answer Key Trigonometric Ratios of Points on the Terminal Side of an Angle Answers 1. (34, 298°) 2. (5 3. (13, 4.39) 4. ( 41,1.79) 5. (4 5,2.03) 6. (10,127°) , 2, 45° ) 4 3 4 5 5 3 sin127° = , cos127° = − , tan127° = − , csc127° = ,sec127° = − , cot127° = − 5 5 3 4 3 4 7. (15, 270°) , sin 270° = −1, cos 270° = 0, tan 270° = und ,csc 270° − 1,sec 270° = und , cot 270° = 0 8. (2 ) 41, 321° , sin 321° = − 4 41 5 41 cot 321° = − 9. , cos 321° = 5 41 4 41 41 , tan 321° = − , csc 321° = − , sec 321° = , 41 5 4 5 4 (8,30°) , 1 3 3 2 3 sin 30° = , cos 30° = , tan 30° = , csc30° = 2,sec30° = , cot 30° = 3 2 2 3 3 10. (6 sin135° = 11. ) 2,135° , 2 2 , cos135° = − , tan135° = −1, csc135° = 2,sec135° = − 2, cot135° = −1 2 2 (9, π ) , sin π = 0,cos π = −1, tan π = 0,csc π = und ,sec π = −1,cot π = und CK-‐12 Algebra II with Trigonometry Concepts 10 Chapter 13 – Trigonometric Ratios 12. 7π ⎞ ⎛ ⎜13 2, ⎟ , 4 ⎠ ⎝ sin 13. ( 7π 2 7π 2 7π 7π 7π 7π =− , cos = , tan = −11, csc = − 2,sec = 2, cot = −1 4 2 4 2 4 4 4 4 ) 13, 0.98 , sin 0.98 = 14. 15. Answer Key 3 13 13 , cos 0.98 = 2 13 13 , tan 0.98 = 3 2 , csc 0.98 = 13 3 , sec 0.98 = 13 2 , cot 0.98 = 2 3 ⎛ 4π ⎞ ⎜14, ⎟ , 3 ⎠ ⎝ sin 4π 3 4π 1 4π 4π 2 3 4π 4π 3 =− , cos = − , tan = 3, csc =− ,sec = −2, cot = 3 2 3 2 3 3 3 3 3 3 (4 5, 2.03 , sin 2.03 = − ) 5 5 , cos 2.03 = − 2 5 5 , tan 2.03 = 1 2 , csc 2.03 = − 5, sec 2.03 = −2 5, cot 2.03 = 2 CK-‐12 Algebra II with Trigonometry Concepts 11 Chapter 13 – Trigonometric Ratios Answer Key Using r and θ to find a Point in the Coordinate Plane 13.11 Answers 1. (10.24, 8.00) 2. (-16.07, 19.15) 3. (16.42, -4.40) 4. (-1.53, 1.29) 5. (2.16, 6.66) 6. (-8.88, 1.45) 7. (2.75, 1.20) 8. (9.01, -4.34) 9. ⎛5 5 3⎞ ⎜⎜ , ⎟⎟ ⎝2 2 ⎠ 10. (3 11. ( −6 12. ( −7,0) 13. ( 0, −11) 14. ( −7, −7 3 ) 15. ⎛ 27 2 27 2 ⎞ , ⎜⎜ − ⎟ 2 2 ⎟⎠ ⎝ 16. ( −20 2, −3 2 3, 6 ) ) 3, −20 ) CK-‐12 Algebra II with Trigonometry Concepts 12 Chapter 13 – Trigonometric Ratios 13.12 Answer Key Law of Sines with AAS and ASA Answers 1. m∠A = 56°, a ≈ 8.7, b ≈ 10.4 2. m∠C = 30°, a ≈ 9.4, b ≈ 6.4 3. m∠A = 65°, c ≈ 5.6, a ≈ 13.6 4. m∠A = 106°, a ≈ 73.8, c ≈ 59.7 5. m∠B = 83°, c ≈ 37.6, b ≈ 41.2 6. m∠C = 33°, b ≈ 16.3, a ≈ 15.2 7. m∠B = 55°, c ≈ 7.7, b ≈ 9.7 8. m∠A = 95°, b ≈ 24.3, c ≈ 11.9 9. m∠C = 102°, a ≈ 7.0, c ≈ 11.7 10. m∠C = 25°, a ≈ 87.2, b ≈ 53.2 11. 79 feet 12. 123.5 meters CK-‐12 Algebra II with Trigonometry Concepts 13 Chapter 13 – Trigonometric Ratios 13.13 Answer Key The Ambiguous Case – SSA Answers 1. 2 triangles 2. 2 triangles 3. 1 triangle 4. No triangle 5. 2 triangles 6. one triangle, m∠B ≈ 39.4° , m∠C ≈ 75.6° and c ≈ 10.7 7. two triangles, m∠B ≈ 61° , m∠C ≈ 78° and c ≈ 13.4 or m∠B ≈ 119° , m∠C ≈ 20° and c ≈ 4.7 8. two triangles, m∠B ≈ 59.6° , m∠C ≈ 87.4° and c ≈ 22 or m∠B ≈ 120.4° , m∠C ≈ 26.6° and c ≈ 9.9 9. one triangle, m∠B ≈ 41° , m∠A ≈ 87° and a ≈ 76 10. no triangle 11. two triangles, m∠B ≈ 78.1° , m∠C ≈ 67.9° and c ≈ 33.1 or m∠B ≈ 101.9° , m∠C ≈ 44.1° and c ≈ 24.9 CK-‐12 Algebra II with Trigonometry Concepts 14 Chapter 13 – Trigonometric Ratios 13.14 Answer Key Area of a Triangle Answers 1. 371 u2 2. 681 u2 3. 35 u2 4. 135 u2 5. 152 u2 6. 94 u2 7. 463 u2 8. 312 u2 9. 1945 u2 10. The two possible measures are 35° and 145° because the sine of an angle and its supplement are equal. 11. 191.5 ft2 12. $97.43 CK-‐12 Algebra II with Trigonometry Concepts 15 Chapter 13 – Trigonometric Ratios 13.15 Answer Key Law of Cosines with SAS (to find the third side) Answers 1. 18.0 2. 23.0 3. 24.9 4. 47.2 5. 15.4 6. 30.9 7. 92.1 8. 15.5 9. 20.1 10. 31.9 11. If cos 90° = 0 , then 12. 30.4 c 2 = a 2 + b2 − 2ab(0) , or c 2 = a 2 + b 2 . CK-‐12 Algebra II with Trigonometry Concepts 16 Chapter 13 – Trigonometric Ratios 13.16 Answer Key Law of Cosines with SSS (to find an angle) Answers 1. 38° 2. 138° 3. 65° 4. 56° 5. 50° 6. 123° 7. 47° 8. 88° 9. 119° 10. 26° 11. 88° 12. 49° CK-‐12 Algebra II with Trigonometry Concepts 17 Chapter 13 – Trigonometric Ratios 13.17 Answer Key Heron’s Formula for the Area of a Triangle and Problem Solving with Trigonometry Answers 1. 0.51 mi 2. 550 ft 3. 3.9 and 7.2 4. 94° 5. 8575 m2 6. 88 in2 7. 1.63 mi; 0.64 mi 8. 73 m2 9. 87 ft 10. 185 ft; 181 ft CK-‐12 Algebra II with Trigonometry Concepts 18