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Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. Math 182: Trigonometry Sample Exam 1 Solutions This practice test will give you an idea of the format and difficulty level of the actual test. Like the actual test, it consists of 27 questions drawn from the problem sets and modified slightly (e.g. by changing the numbers). Any problem on the problem sets can appear on the actual test; it is not enough just to know how to solve the problems on this practice test. No partial credit will be given for wrong answers. You will be graded on the best 25 of the 27 questions, so you can get up to two answers wrong without penalty. You cannot score more than 100% on the test. Read the instructions carefully and follow them exactly. Your answers on the answer page must exactly match the solutions, or they will be graded as wrong. If the question asks that you round the answer to a certain number of decimal places, you must do so. Your answer will be considered wrong if it is rounded incorrectly, or to the wrong number of decimal places. Unless you’re told to round your answer, it should be exact. Some things that can cause you to lose credit for an exact answer are: 15 3 ● Failure to simplify fractions. is wrong; is correct. 20 4 2 2 3 ● Failure to rationalize denominators. is wrong; is correct. 3 3 ● Failure to simplify radicals. 18 is wrong; 3 2 is correct. ● Using decimal approximations instead of exact answers. 1.0472 is wrong; π is correct. 3 On problems where the answer involves units of measure (e.g. “23 ft”), you won’t lose credit for not including the units. However, it’s a good idea to do so. Your answer must be in the correct units: if, for example, the correct answer is “30°”, then “π/6” will be graded as wrong. Your answers must appear in the correct format on the answer sheet. If no answer or a wrong answer appears there, the grader will not check the page with the question to see if you’ve answered it correctly there. That means it’s a good idea to double-check at the end of the test and make sure that you’ve copied the answers correctly and in the right places on the answer sheet. Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. Math 182: Trigonometry Sample Exam 1 Name Date Score Grader This exam consists of 27 multiple-choice and short-answer questions. There will be no partial credit for wrong answers. You will be graded on the best 25 of the 27, so you can get two questions wrong without penalty. You cannot get a higher grade than 100% on this exam. Write only your answers in the spaces provided on this page. Circle the correct answers for multiple-choice questions. Do not do your work on this page. Write your answers in exactly the format that the question asks for. If, for example, you round to the wrong number of decimal places, or fail to rationalize denominators or simplify fractions and radicals in exact solutions, your answer will be graded as wrong. Unless otherwise indicated, your answers should be exact. Rationalize all denominators and simplify all fractions and radicals. Your answer must be in the correct units of measure. If, for example, the problem asks for an angle in degrees, then an answer given in radians would be considered wrong. Your grade will be based on the answers that you write on this page. If you have a wrong answer or no answer on this page, the grader will not look at the page with the question to see if the correct answer appears there. Illegible or ambiguous answers will be graded as wrong. You are responsible for copying your answers clearly, correctly, and in the appropriate blanks. You must show your work on the page with the question. Credit will not be given for lucky guesses. 1. 4.5 cm 8. 2/3 15. 2.1790 rad 22. b 2. 66° 9. 2 53 53 16. 59.74° 23. e 3. 18.3503 10. 2π/9 rad 17. 1/2 24. e 4. arccos(5/7) 11. 13.5788 18. 0.4797 rad 25. a 5. b 12. c 19. 113 ft 26. a 6. 349 ft 13. 0.4447 rad 20. b 27. 1.9443 rad 7. III 14. d 21. a Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 1. The radius of a circle is 15 cm. A central angle measures 0.3 radians. What is the length of the arc intercepted by the angle? Your answer should be exact. Use the formula: s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. Then s = (15)(0.3) = 4.5 cm. 2. What is the complement of 24°? The complement of θ° is 90 – θ. In this case, it’s 90 – 24 = 66°. 3. In the figure at right, the triangle ABC has angles A = 0.61 rad and B = 1.13 rad, and side c = 20. Find the length of side b. Round your answer to four decimal places. This is an ASA problem. Start by calculting C = π – (A + B) = 1.4016. (Store the value of C in your calculator; do not write down “1.4016” and re-enter it, or you risk round-off error.) Now use the law of sines: 20 b 20 sin(1.13) ; so b = = 18.3503. = sin(1.13) sin C sin C 4. For the right triangle at right, express θ as an arc function, e.g. θ = arcsin(3/4). Do not calculate the actual value of θ. cos θ = 5/7 (and θ is acute); so θ = arccos(5/7) C b A a c B 7 θ 5 5. An angle of 2π/5 radians is: (a) straight (b) acute (c) laminar (d) right (e) obtuse (f) haploid 2π/5 < π/2; so the angle is acute. The correct answer is (b). Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 6. You want to know the width of a river. You begin by standing directly across from a conspicuous tree on the opposite bank. You then walk 170 feet straight downstream. From the new point, the tree is at an angle of θ = 64° to the upstream direction. How wide is the river? Round your answer to the nearest foot. y Let y be the width of the river. Then tan 64° = , so 170 y = 170 tan 64° = 349 ft θ 7. In which quadrant would you find 213°? Circle the correct answer on the answer page. 180° < 213° < 270°; so 213° is in the third quadrant. The correct answer is III. 8. Fill in the blank to make the statement true. Assume θ is in the first quadrant. cos θ = _________ if and only if θ = arccos(2/3) 2/3; this comes straight from the definition of an arc cosine 9. In the right triangle at right, what is sin A? Your answer should be exact. 2 A Use Pythagoras to find the hypotenuse: r = 7 2 + 2 2 = 53 . Then 7 2 53 . Don’t forget to rationalize the denominator: otherwise, your answer will be 53 graded as wrong. sin A = 10. An angle measures 40°. What is its measure in radians? Your answer should be exact. 180° = π radians; so to convert from degrees to radians, multiply by π 180 . Thus 40π 2π radians. Your answer will be counted as wrong if you don’t simplify the = 180 9 fraction, or if you use a decimal approximation like 0.698132. 40° = Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 11. In the figure at right, the triangle ABC has angle A = 53° and sides b = 17 and c = 10. Find the length of side a. Round your answer to four decimal places. This is an SAS situation. By the law of cosines, a 2 = 17 2 + 10 2 − 4(17)(10) cos 53° ; so a = 13.5788. B a c b A C 12. Which function does the graph below represent? Angles are measured in degrees. (a) y = sin x (b) y = cot x (b) y = cos x (e) y = sec x (c) y = tan x (f) y = csc x 10 8 6 4 2 0 -120 -60 0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 -2 -4 -6 -8 -10 The graphs of sin x and cos x oscillate back and forth between -1 and 1, so this isn’t one of those. The values of sec x and cos x can never be zero; they must always be in the range [1,∞) or (-∞,-1]. Thus this graph is either tan x or cot x. It goes through the point (0,0). That’s consistent with y = tan x, since tan(0) = 0; but inconsistent with y = cot x, since cot(0) is undefined. Hence the correct answer is (c). Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 13. Your friend’s house is 8520 feet away from yours. An airplane flies over your friend’s house at an elevation of 4060 feet. What is the angle of elevation of the airplane, as seen from your house? Round your answer to the nearest 0.0001 radians. 4060 Let θ be the angle of elevation. Then tan θ = . Since θ is an 8520 4060 acute angle, θ = tan −1 = 0.4447 rad. 8520 θ 14. If cos θ > 0, which quadrant could θ be in? (a) I or II (b) I or III (c) I or IV (d) II or III (e) II or IV (f) III or IV cos θ is negative when θ is in the II or III quadrant; so the correct answer is (d). 15. In the figure at right, the triangle ABC has sides a = 12, b = 17, and c = 7. What is the measure of angle B? Round your answer to the nearest 0.0001 radian. This is an SSS situation. By the law of cosines, 12 2 + 7 2 − 17 2 . Thus B = 2.1790 rad. cos B = 2(12)(7) B a c A 16. In the right triangle at right, calculate the angle θ. Round your answer to the nearest 0.01°. 180 12 tan −1 = 59.74°. tan θ = 12/7; so θ = π 7 C b 12 θ 7 Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 17. If θ = 5π/3 radians, what is cos θ? Do not use a calculator. Your answer should be exact. θ is in the fourth quadrant, so the reference angle is 2π – θ = π/3 rad. The cosine of π/3 is 1/2. Since θ is in the fourth quadrant, its cosine is positive; so cos θ = 1/2. 18. In the right triangle at right, calculate the angle A. Round your answer to the nearest 0.0001 rad. sin A = 6/13; so A = arcsin(6/13) = 0.4797 rad. B 13 6 C A 19. You are flying a kite on a windy day. The string is 180 feet long, and the wind is strong enough to stretch it straight. A friend who’s standing 140 feet from you tells you that the kite is directly over him. How high above the ground is the kite? Round your answer to the nearest foot. You, the kite, and the friend form a right triangle with hypotenuse r = 180 and with side x = 140. You want the third side: the height h of the kite. By Pythagoras, h = 180 2 − 140 2 = 113 ft. 20. Find the distance between the points (-1,4) and (3,2). (a) 2 2 (d) 10 (b) 2 5 (e) 6 (c) 26 (f) 4 The distance between points ( x1 , y1 ) and ( x2 , y 2 ) is: d = (x2 − x1 )2 + ( y 2 − y1 )2 . our numbers gives d = 4 2 + 2 2 = 20 = 2 5 . Hence the correct answer is (b). Plugging in Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 21. What is the amplitude of the function: y = 6 cos (5x − 7)? Angles are measured in radians. (a) 6 (b) 5 (c) 7 (d) 12 (e) 10 (f) 14 In a function of the form y = a cos(bx − c ) + d , the amplitude is a. In this problem, it’s 6. Hence the correct answer is (a). 22. A boat is bobbing up and down on the waves. The depth of water z in meters below the boat as a function of time t in seconds is: z = 0.8 sin (bt ) + 8.5 , where angles are measured in degrees. If two consecutive high points are separated by 10 sec, what is b? (a) 6 (b) 36 (c) 10 (d) 1/6 (e) 1/36 (f) 1/10 In a function of the form y = a sin(bx − c) + d , with angles measured in degrees, the period is 360 . Two consecutive maxima of the function are separated by one period; so in this T= b 360 problem, T = 10 sec. Hence b = = 36. The correct answer is (b). 10 23. If θ is in the fourth quadrant, and cosθ = (a) 7 4 (d) − (b) 7 4 11 , what is sin θ? 4 5 4 (e) − 5 4 We can use the identity sin 2 θ + cos 2 θ = 1 ; so sin 2 θ = 1 − quadrant, sin θ is negative; so sin θ = − (c) 3 4 (f) − 3 4 11 5 = . Since θ is in the fourth 16 16 5 . The correct answer is (e). 4 Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 24. What function does the graph below show? x is in degrees. (a) y = sin(3 x − 90) (b) y = sin( 2 x) − 5 (c) y = 3 sin(2 x) − 5 x x (d) y = 5 sin + 60 (e) y = 3 sin − 2 (f) y = −6 sin(3 x) + 1 2 2 1.5 1 0.5 -120 0 -60-0.5 0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 We want to find a function of the form y = a sin(bx − c) + d that matches this graph. The easiest thing to check is the amplitude a. The graph ranges between a maximum of 1 and a minimum of -5; its amplitude is half the difference between those, so a = 3. That immediately rules out all of the choices except (c) and (e). Next, look at the vertical translation d. The midpoint between the maximum and minimum is −2; so the graph has been translated down by 2 units, and d = -2. That excludes (c), so (e) is the only choice left. Just to be safe, we’ll check the period and the horizontal translation. The graph rises through the midpoint at (0,-2) and again at (720,-2), so the period is T = 720. When x is measured in degrees, bT = 360; so b = 1/2. That’s consistent with (e) but not with (c). The graph rises through the midpoint at (0,-2), which suggests that c = 0. The graph of (a) would rise through the midline at x = 30; the graph of (d) would rise through the midline at x = 120. The correct answer is (e). Hackernotes, dedicated to the Ubuntu community © Wayne Hacker 2007. All rights reserved. 25. tan θ = ? sin θ sin θ sin θ (a) (b) (c) cosθ cscθ secθ cosθ cosθ cosθ (d) (e) (f) sin θ cscθ secθ This is a basic trig identity that you should know by memory. It’s (a). 26. If angles are measured in radians, sin θ = ? π (a) cos − θ (b) cos(π − θ ) (c) 1 − cos θ 2 π (d) − cos − θ (e) − cos(π − θ ) (f) 1 − cos 2 θ 2 The sine of an angle is the cosine of its complement. Hence: (a). 27. In the triangle ABC shown at right, A = 0.65 rad; a = 13; c = 20; and C is obtuse. Calculate the value of C. Round your answer to the nearest 0.0001 radian. This is an SSA situation. From the law of sines, sin(0.65) sin C 20 sin(0.65) ; so sin C = . We want C to be obtuse; = 13 20 13 20 sin 0.65 so if we let θ = sin −1 , then C = π − θ = 1.9443 rad. 13 B c a A b C
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