Unit 5 assaignment- Precalculus

450 CHAPTER 5 Analytic Trigonometry 52. (a) Prove the identity: b 2 + c2 – a 2 cos A = . a 2abc (a) How fast is each ship traveling? (Express your answer in knots, which are nautical miles per hour.) (b) What is the angle of intersection of the courses of the two ships? (b) Prove the (tougher) identity: 2 2 2 cos A cos B a + b + c cos C + = + . a c b 2abc (c) How far apart are the ships at 12:00 noon if they maintain the same courses and speeds? [Hint: Use the identity in part (a), along with its other variations.] Extending the Ideas 53. Navigation Two ships leave a common port at 8:00 A.M. and travel at a constant rate of speed. Each ship keeps a log showing its distance from port and its distance from the other ship. Portions of the logs from later that morning for both ships are shown in the following tables. Naut mi from Naut mi from Time port ship B 9:00 10:00 15.1 30.2 8.7 17.3 54. Prove that the area of a triangle can be found with the formula ¢ Area = Naut mi from Naut mi from Time port ship A 9:00 11:00 12.4 37.2 8.7 26.0 a 2 sin B sin C . 2 sin A 55. A segment of a circle is the region enclosed between a chord of a circle and the arc intercepted by the chord. Find the area of a segment intercepted by a 7-inch chord in a circle of radius 5 inches. 5 7 5 CHAPTER 5 Key Ideas Properties, Theorems, and Formulas Reciprocal Identities 404 Quotient Identities 404 Pythagorean Identities 405 Cofunction Identities 406 Odd-Even Identities 407 Sum/Difference Identities 422–424 Double-Angle Identities 428 Power-Reducing Identities 428 Half-Angle Identities 430 Law of Sines 434 Law of Cosines 442 Triangle Area 444 Heron’s Formula 445 Procedures Strategies for Proving an Identity 413–415 CHAPTER 5 Review Exercises Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. The collection of exercises marked in red could be used as a chapter test. In Exercises 1 and 2, write the expression as the sine, cosine, or tangent of an angle. 1. 2 sin 100° cos 100° 2 tan 40° 2. 1 – tan2 40° In Exercises 3 and 4, simplify the expression to a single term. Support your answer graphically. 3. 11 – 2 sin2 u22 + 4 sin2 u cos2 u 4. 1 – 4 sin2 x cos2 x In Exercises 5–22, prove the identity. 5. cos 3x = 4 cos3 x – 3 cos x 6. cos2 2x – cos2 x = sin2 x – sin2 2x 7. tan2 x – sin2 x = sin2 x tan2 x SECTION 5 8. 2 sin u cos3 u + 2 sin3 u cos u = sin 2u 9. csc x – cos x cot x = sin x 10. tan u + sin u u = cos2 a b 2 tan u 2 11. 1 + tan u 1 + cot u + = 0 1 – tan u 1 – cot u 12. sin 3u = 3 cos2 u sin u – sin3 u 39. 2 cos x = 1 40. sin 3x = sin x 41. sin2 x – 2 sin x – 3 = 0 42. cos 2t = cos t 43. sin 1cos x2 = 1 44. cos 2x + 5 cos x = 2 In Exercises 45–48, solve the inequality. Use any method, but give exact answers. 45. 2 cos 2x 7 1 for 0 … x 6 2p tan3 g – cot 3 g 47. 2 cos x 6 1 for 0 … x 6 2p p p 48. tan x 6 sin x for – 6 x 6 2 2 15. 16. 17. tan2 g + csc2 g 46. sin 2x 7 2 cos x for 0 6 x … 2p = tan g – cot g cos f sin f + = cos f + sin f 1 – tan f 1 – cot f cos 1- z2 sec 1- z2 + tan 1- z2 = 1 + sin z ƒ cos g ƒ 1 – sin g 1 – cos y 1 – cos y = = 18. A A 1 + cos y 1 + sin g 1 + sin g ƒ sin y ƒ 19. tan a u + 451 In Exercises 39– 44, find all solutions in the interval 30, 2p2 without using a calculator. Give exact answers. t 1 + sec t 13. cos2 a b = 2 2 sec t 14. Review Exercises In Exercises 49 and 50, find an equivalent equation of the form y = a sin 1bx + c2. Support your work graphically. 49. y = 3 sin 3x + 4 cos 3x 50. y = 5 sin 2x – 12 cos 2x In Exercises 51–58, solve ¢ABC. A b tan u – 1 3p b = 4 1 + tan u 1 sin 4g = sin g cos3 g – cos g sin3 g 4 1 21. tan b = csc b – cot b 2 2t 1 , -1 6 t 6 1 22. arctan t = arctan 2 1 – t2 20. c C a 51. A = 79°, B B = 33°, a = 7 52. a = 5, b = 8, B = 110° 53. a = 8, b = 3, B = 30° 54. a = 14.7, A = 29.3°, C = 33° 55. A = 34°, B = 74°, c = 5 56. c = 41, A = 22.9°, C = 55.1° 23. sec x – sin x tan x = cos x 57. a = 5, b = 7, 24. 1sin2 a – cos2 a21tan2 a + 12 = tan2 a – 1 58. A = 85°, In Exercises 23 and 24, use a grapher to conjecture whether the equation is likely to be an identity. Confirm your conjecture. In Exercises 25–28, write the expression in terms of sin x and cos x only. a = 6, c = 6 b = 4 In Exercises 59 and 60, find the area of ¢ABC. 59. a = 3, b = 5, c = 6 b = 6, C = 50° 25. sin 3x + cos 3x 26. sin 2x + cos 3x 60. a = 10, 27. cos2 2x – sin 2x 28. sin 3x – 3 sin 2x 61. If a = 12 and B = 28°, determine the values of b that will produce the indicated number of triangles: In Exercises 29–34, find the general solution without using a calculator. Give exact answers. 13 29. sin 2x = 0.5 30. cos x = 2 (a) Two (b) One (c) Zero 31. tan x = -1 32. 2 sin-1 x = 12 62. Surveying a Canyon Two markers A and B on the same side of a canyon rim are 80 ft apart, as shown in the figure. A hiker is located across the rim at point C. A surveyor determines that ∠ BAC = 70° and ∠ ABC = 65°. 33. tan-1 x = 1 34. 2 cos 2x = 1 (a) What is the distance between the hiker and point A? In Exercises 35–38, solve the equation graphically. Find all solutions in the interval 30, 2p2. (b) What is the distance between the two canyon rims? (Assume they are parallel.) 35. sin2 x – 3 cos x = – 0.5 A 36. cos3 x – 2 sin x – 0.7 = 0 37. sin4 x + x 2 = 2 38. sin 2x = x 3 – 5x 2 + 5x + 1 C 70° 80 ft 65° B 452 CHAPTER 5 Analytic Trigonometry 63. Altitude A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon? (a) Graph the function y = S1u2. (b) What value of u gives the minimum surface area? (Note: This answer is quite close to the observed angle in nature.) (c) What is the minimum surface area? 33° A 37° B 1.75 mi 64. Finding Distance In order to determine the distance between two points A and B on opposite sides of a lake, a surveyor chooses a point C that is 900 ft from A and 225 ft from B, as shown in the figure. If the measure of the angle at C is 70°, find the distance between A and B. 225 ft 70. Finding Extremum Values The graph of 70° 900 ft C y = cos x – 65. Finding Radian Measure Find the radian measure of the largest angle of the triangle whose sides have lengths 8, 9, and 10. 66. Finding a Parallelogram A parallelogram has sides of 15 and 24 ft, and an angle of 40°. Find the diagonals. 67. Maximizing Area A trapezoid is inscribed in the upper half of a unit circle, as shown y in the figure. (a) Write the area of the trapezoid as a function of u. x2 + y2 = 1 (b) Find the value of u that maximizes the area of the trapezoid and the maximum area. (–1, 0) (x, y) θ x (1, 0) 68. Beehive Cells A single cell in a Rear beehive is a regular hexagonal prism of cell open at the front with a trihedral cut at the back. Trihedral refers to a vertex formed by three faces of a polyhedron. It can be shown that the surface area of a cell is given by S1u2 = 6ab + Trihedral angle ␪ a 3 2 13 b a – cot u + b, 2 sin u b where u is the angle between the axis of the prism and one of the back faces, a is the depth of the prism, and b is the length of the hexagonal front. Assume a = 1.75 in. and b = 0.65 in. (a) Assuming that the Earth is spherical with a radius of 4000 mi, write h as a function of u. (b) Approximate u for a satellite 200 mi above the surface of the Earth. B A 69. Cable Television Coverage A cable broadcast satellite S orbits a planet at a height h (in miles) above the Earth’s surface, as shown in the figure. Earth The two lines from S are tangent to the Earth’s surface. The part S h of the Earth’s surface that is in θ the broadcast area of the satellite is determined by the central r angle u indicated in the figure. b Front of cell 1 1 cos 2x + cos 3x 2 3 is shown in the figure. The x-values that correspond to local maximum and minimum points are solutions of the equation sin x – sin 2x + sin 3x = 0. Solve this equation algebraically, and support your solution using the graph of y. [–2π , 2π ] by [–2, 2] 71. Using Trigonometry in Geometry A regular hexagon whose sides are 16 cm is inscribed in a circle. Find the area inside the circle and outside the hexagon. 72. Using Trigonometry in Geometry A circle is inscribed in a regular pentagon whose sides are 12 cm. Find the area inside the pentagon and outside the circle. 73. Using Trigonometry in Geometry A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15° is cut from the wheel, find the volume of the cheese wedge. 74. Product-to-Sum Formulas Prove the following identities, which are called the product-to-sum formulas. 1 (a) sin u sin v = 1cos 1u – v2 – cos 1u + v22 2 1 (b) cos u cos v = 1cos 1u – v2 + cos 1u + v22 2 1 (c) sin u sin v = 1sin 1u + v2 + sin 1u – v22 2 75. Sum-to-Product Formulas Use the product-to-sum formulas in Exercise 74 to prove the following identities, which are called the sum-to-product formulas. SECTION 5 u + v u – v cos 2 2 u – v u + v (b) sin u – sin v = 2 sin cos 2 2 u – v u + v (c) cos u + cos v = 2 cos cos 2 2 u + v u – v (d) cos u – cos v = – 2 sin sin 2 2 77. An Interesting Fact About (sin A) /a The ratio 1sin A2/a that shows up in the Law of Sines shows up another way in the geometry of ¢ABC : It is the reciprocal of the radius of the B A′ circumscribed circle. (a) sin u + sin v = 2 sin (a) Let ¢ABC be circumscribed as shown in the diagram, and construct diameter CA¿ . Explain why ∠ A¿BC is a right angle. 76. Catching Students Faking Data Carmen and Pat both need to Mirror C make up a missed physics lab. ␪ They are to measure the total x x distance 12×2 traveled by a A B beam of light from point A 24⬙ to point B and record it in 20° increments of u as they adjust the mirror 1C2 upward vertically. They report the following measurements. However, only one of the students actually did the lab; the other skipped it and faked the data. Who faked the data, and how can you tell? CARMEN 453 Review Exercises PAT u 2x u 2x 160° 140° 120° 100° 80° 60° 40° 20° 24.4– 25.6– 28.0– 31.2– 37.6– 48.0– 70.4– 138.4– 160° 140° 120° 100° 80° 60° 40° 20° 24.5– 25.2– 26.4– 30.4– 35.2– 48.0– 84.0– 138.4– C A (b) Explain why ∠ A¿ and ∠ A are congruent. (c) If a, b, and c are the sides opposite angles A, B, and C as usual, explain why sin A¿ = a/d, where d is the diameter of the circle. (d) Finally, explain why 1sin A2/a = 1/d. (e) Do 1sin B2/b and 1sin C2/c also equal 1/d? Why? 454 CHAPTER 5 Analytic Trigonometry CHAPTER 5 Project Modeling the Illumination of the Moon From the Earth, the Moon appears to be a circular disk in the sky that is illuminated to varying degrees by direct sunlight. During each lunar orbit the Moon varies from a status of being a new Moon with no visible illumination to that of a full Moon, which is fully illuminated by direct sunlight. The U.S. Naval Observatory has developed a mathematical model to find the fraction of the Moon’s visible disk that is illuminated by the Sun. The data in the table below (obtained from the U.S. Naval Observatory Web site, http://aa.usno.navy.mil/, Astronomical Applications Department) show the fraction of the Moon illuminated at midnight for each day in January 2010. Fraction of the Moon Illuminated, January 2005 Day # Fraction illuminated Day # Fraction illuminated Day # Fraction illuminated 1 2 3 4 5 6 7 8 1.00 0.97 0.92 0.84 0.74 0.64 0.53 0.42 9 10 11 12 13 14 15 16 0.32 0.23 0.15 0.09 0.04 0.01 0.00 0.01 17 18 19 20 21 22 23 24 0.03 0.07 0.13 0.20 0.28 0.38 0.48 0.58 Day # Fraction illuminated 25 26 27 28 29 30 31 0.68 0.78 0.87 0.94 0.98 1.00 0.99 EXPLORATIONS 1. Enter the data in the table above into your grapher or com- 5. Find values for a, b, h, and k so the equation puter. Create a scatter plot of the data. 2. Find values for a, b, h, and k so the equation y = a cos 1b1x – h22 + k models the data in the data plot. 3. Verify graphically the cofunction identity sin 1p/2 – u2 = cos u by substituting 1p/2 – u2 for u in the model above and using sine instead of cosine. 1Note u = b1x – h2.2 Observe how well this new model fits the data. 4. Verify graphically the odd-even identity cos 1u2 = cos 1-u2 for the model in #2 by substituting -u for u and observing how well the graph fits the data. 6. Verify graphically the cofunction identity cos 1p/2 – u2 = y = a sin 1b1x – h22 + k fits the data in the table. sin u by substituting 1p/2 – u2 for u in the model above and using cosine rather than sine. 1Note u = b1x – h2.2 Observe the fit of this model to the data. 7. Verify graphically the odd-even identity sin 1 -u2 = – sin 1u2 for the model in #5 by substituting – u for u and graphing – a sin 1-u2 + k. How does this model compare to the original one?

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