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1 Practice Test 2 Part A Chap 1 Sections 5,6,7,8 ( ) Question Description This is one of two practice tests to help you prepare for Test 2. The other is Practice Test 2 Part B Chap 2 Sections 1,2,3. 1. Question Details LarTrig [ ] Find the period and amplitude. y = 1 9 sin 2πx period amplitude 2. Question Details LarTrig [ ] Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. f(x) = sin x g(x) = sin(x 5π) The graph of g has an amplitude 5π times that of the graph of f. The graph of g is a vertical shift of the graph of f 5π units downward. The graph of g is identical to the graph of f. The graph of g has a period 5π units longer than the period of the graph of f. The graph of g is a horizontal shift of the graph of f 5π units to the right. 3. Question Details LarTrig [ ] Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. f(x) = cos 5x g(x) = cos 5x The graph of g is a reflection of the graph of f in the xaxis. The graph of g is a vertical shift of the graph of f 5 units down. The graph of g is a reflection of the graph of f in the yaxis. The graph of g has an amplitude of 5 times that of the amplitude of the graph of f. The graph of g has a period of 5 times as long as the period of the graph of f. 1 of 35 10/20/ :54 AM
2 4. Question Details LarTrig [ ] Describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. f(x) = sin 8x g(x) = 4 + sin 8x The graph of g is a horizontal shift on the graph of f to the left by 4 units. The graph of g is a vertical shift on the graph of f downward by 4 units. The graph of g is a horizontal shift on the graph of f to the right by 4 units. The graph of g has a period 8 units shorter than the period of the graph of f. The graph of g is a vertical shift on the graph of f upward by 4 units. 2 of 35 10/20/ :54 AM
3 5. Question Details LarTrig [ ] Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.) f(x) = 7 cos 7x g(x) = cos 14x Solution or Explanation f(x) = 7 cos 7x 2π 2π Period: = b 7 Amplitude: 7 Symmetry: yaxis Key points: Maximum Intercept Minimum Intercept Maximum π π (0, 7), 0, 7 (3/14 π, 0) (2/7 π, 7) Since g(x) = cos(14x) = f(2x), the graph of g(x) is the graph of f(x), but 7 i) shrunk horizontally by a factor of 2, 1 ii) shrunk vertically by a factor of, and 7 iii) reflected about the xaxis. Generate key points for the graph of g(x) by i) dividing the xcoordinate of each key point of f(x) by 2, and ii) dividing the ycoordinate of each key point of f(x) by 7 3 of 35 10/20/ :54 AM
4 6. Question Details LarTrig [ ] Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.) f(x) = cos x g(x) = 1 + cos x 4 of 35 10/20/ :54 AM
5 7. Question Details LarTrig MI. [ ] Sketch the graphs of f and g in the same coordinate plane. (Include two full periods.) f(x) = 8 sin πx g(x) = 8 sin πx 3 Solution or Explanation f(x) = 8 sin πx 2π 2π Period: = = 2 b π Amplitude: 8 Symmetry: origin Key points: Intercept Maximum Intercept Minimum Intercept 1 3 (0, 0), 8 (1, 0), 8 (2, 0) 2 2 Since g(x) = 8 sin πx 3 = f(x) 3, the graph of g(x) is the graph of f(x), but translated downward by three units. Generate key points for the graph of g(x) by subtracting 3 from the ycoordinate of each key point of f(x). 5 of 35 10/20/ :54 AM
6 8. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 4 sin 4πx 3 6 of 35 10/20/ :54 AM
7 9. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 3 cos(x + π) 3 7 of 35 10/20/ :54 AM
8 10. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 3 cos(6x + π) 8 of 35 10/20/ :54 AM
9 11. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) 3 x y = cos 4 2 π Question Details LarTrig [ ] Fill in the blank. The domain of y = cot x is all real numbers such that. x nπ, where n is an integer x 0 x π x = nπ, where n is an integer x nπ, 2 where n is an odd integer 9 of 35 10/20/ :54 AM
10 13. Question Details LarTrig [ ] Fill in the blank. The range of y = sec x is. [ 1, 1] (, 1] [1, ) (, ) ( 1, 1) (, 1) (1, ) 10 of 35 10/20/ :54 AM
11 14. Question Details LarTrig [ ] Match the function with its graph. y = sec 3x State the period of the function. 11 of 35 10/20/ :54 AM
12 15. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 1 3 sec x 12 of 35 10/20/ :54 AM
13 16. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 3 sec 3x + 3 Solution or Explanation y = 3 sec 3x + 3 2π 2π Period: = 3 3 Two consecutive asymptotes: π π x =, x = 6 6 x π π y of 35 10/20/ :54 AM
14 17. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 3 cot 4x 14 of 35 10/20/ :54 AM
15 18. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = tan(x + π) 15 of 35 10/20/ :54 AM
16 19. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = 4 csc(x π) 16 of 35 10/20/ :54 AM
17 20. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) y = sec πx of 35 10/20/ :54 AM
18 21. Question Details LarTrig [ ] Sketch the graph of the function. (Include two full periods.) 1 y = csc x + 3 π 3 18 of 35 10/20/ :54 AM
19 22. Question Details LarTrig [ ] Use a graphing utility to graph the function. (Include two full periods.) 1 πx y = sec π Question Details LarTrig [ ] Evaluate the expression without using a calculator. (Enter your answer in radians.) arctan Question Details LarTrig [ ] Evaluate the expression without using a calculator. (Enter your answer in radians.) arccos 1 19 of 35 10/20/ :54 AM
20 25. Question Details LarTrig [ ] Evaluate the expression without using a calculator. (Enter your answer in radians.) arcsin Question Details LarTrig [ ] Evaluate the expression without using a calculator. (Enter your answer in radians.) arctan Question Details LarTrig [ ] Evaluate the expression without using a calculator. (Enter your answer in radians.) arcsin Question Details LarTrig [ ] Use a calculator to evaluate the expression. Round your result to two decimal places. (Enter your answer in radians.) arcsin of 35 10/20/ :54 AM
21 29. Question Details LarTrig [ ] Determine the missing coordinates of the points on the graph of the function. (x 1, y 1 ) = 1, (x 2, y 2 ) =, π 6 (x 3, y 3 ) =, π 3 21 of 35 10/20/ :54 AM
22 30. Question Details LarTrig [ ] Determine the missing coordinates of the points on the graph of the function. x 1, y 1 = 1, 2 x 2, y 2 = 0, x 3, y 3 =, π Question Details LarTrig [ ] Use an inverse trigonometric function to write θ as a function of x. θ = 22 of 35 10/20/ :54 AM
23 32. Question Details LarTrig [ ] Use an inverse trigonometric function to write θ as a function of x. θ = 33. Question Details LarTrig [ ] Use an inverse trigonometric function to write θ as a function of x. θ = Solution or Explanation tan(θ) = x θ = arctan x of 35 10/20/ :54 AM
24 34. Question Details LarTrig [ ] Use the properties of inverse trigonometric functions to evaluate the expression. sin(arcsin 0.8) 35. Question Details LarTrig [ ] Use the properties of inverse trigonometric functions to evaluate the expression. tan(arctan 52) 36. Question Details LarTrig [ ] Use the properties of inverse trigonometric functions to evaluate the expression. cos[arccos( 0.5)] 37. Question Details LarTrig [ ] Use the properties of inverse trigonometric functions to evaluate the expression. arcsin(sin 6π) 38. Question Details LarTrig [ ] Use the properties of inverse trigonometric functions to evaluate the expression. (Enter your answer in radians.) arccos cos 11π 2 Solution or Explanation 11π π arccos cos = arccos 0 = π Note: is not in the range of the arccosine function of 35 10/20/ :54 AM
25 39. Question Details LarTrig [ ] Find the exact value of the expression. (Hint: Sketch a right triangle.) sin arctan Question Details LarTrig MI. [ ] Find the exact value of the expression. (Hint: Sketch a right triangle.) sec arcsin 4 5 5/3 Solution or Explanation 4 Let u = arcsin, 5 4 π sin(u) =, 0 < u <, sec arcsin = sec(u) = Question Details LarTrig [ ] Find the exact value of the expression. (Hint: Sketch a right triangle.) cos(tan 1 3) 25 of 35 10/20/ :54 AM
26 42. Question Details LarTrig [ ] Find the exact value of the expression. (Hint: Sketch a right triangle.) sin cos Question Details LarTrig [ ] Fill in the blank. The number of cycles per second of a point in simple harmonic motion is its Select frequency. 44. Question Details LarTrig [ ] Solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. A = 30, b = 9 B = 60 C = 90 a = 5.20 c = Question Details LarTrig [ ] Fill in the blank. A(n) Select bearing measures the acute angle a path or line of sight makes with a fixed northsouth line. Solution or Explanation bearing 46. Question Details LarTrig [ ] Fill in the blank. A point that moves on a coordinate line is said to be in simple Select when its distance d from the origin at time t is given by either d = a sin ωt or d = a cos ωt. harmonic motion 26 of 35 10/20/ :54 AM
27 47. Question Details LarTrig [ ] Fill in the blank. The time for one complete cycle of a point in simple harmonic motion is its Select period. 48. Question Details LarTrig MI. [ ] Solve the right triangle shown in the figure. Round your answers to two decimal places. B = 62, c = 17 A = 28 C = 90 a = 7.98 b = Solution or Explanation Given: B = 62, c = 17 A = 90 B = = 28 a cos(b) = a = c cos(b) = 17 cos(62 ) 7.98 c b sin(b) = b = c sin(b) c = 17 sin(62 ) B = 62 c = of 35 10/20/ :54 AM
28 49. Question Details LarTrig [ ] Solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. B = 75, b = 24 A = 15 C = 90 a = 6.43 c = of 35 10/20/ :54 AM
29 50. Question Details LarTrig [ ] Solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. a = 24, c = 32 A = B = C = 90 b = Solution or Explanation Given: a = 24, c = 32 a a sin(a) = A = arcsin c c 24 = arcsin a a 24 cos(b) = B = arccos = arccos c c 32 b = c 2 a 2 = = a = 24 c = of 35 10/20/ :54 AM
30 51. Question Details LarTrig [ ] Solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places. b = 18, c = 45 A = B = C = 90 a = Question Details LarTrig [ ] Find the altitude of the isosceles triangle shown in the figure. Round your answer to two decimal places. θ = 45, b = Question Details LarTrig [ ] The sun is 25 above the horizon. Find the length of a shadow cast by a building that is 100 feet tall (see figure). (Round your answer to two decimal places.) ft 30 of 35 10/20/ :54 AM
31 54. Question Details LarTrig MI. [ ] The sun is 24 above the horizon. Find the length of a shadow cast by a park statue that is 20 feet tall. (Round your answer to one decimal place.) 44.9 ft Solution or Explanation 20 tan 24 = x 20 x = tan ft 55. Question Details LarTrig MI.SA. [ ] This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise The sun is 18 above the horizon. Find the length of a shadow cast by a park statue that is 30 feet tall. 56. Question Details LarTrig [ ] A ladder that is 30 feet long leans against the side of a house. The angle of elevation of the ladder is 80. Find the height from the top of the ladder to the ground. (Round your answer to one decimal place.) 29.5 ft 57. Question Details LarTrig [ ] At a point 45 feet from the base of a church, the angles of elevation to the bottom of the steeple and the top of the steeple are 35 and 48 30', respectively. Find the height of the steeple. (Round your answer to one decimal place.) 19.4 ft 31 of 35 10/20/ :54 AM
32 58. Question Details LarTrig [ ] An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to the ships are β = 2 and θ = 3.5 (see figure). How far apart are the ships? (Round your answer to one decimal place.) ft 59. Question Details LarTrig [ ] A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. The angles of depression to the towns are 21 and 51 (see figure). How far apart are the towns? (Round your answer to one decimal place.) 18.0 km Question Details LarTrig [ ] You observe a plane approaching overhead and assume that its speed is 650 miles per hour. The angle of elevation of the plane is 15 at one time and 59 one minute later. Approximate the altitude of the plane. (Round your answer to two decimal places.) 3.46 mi 61. Question Details LarTrig [ ] During takeoff, an airplane's angle of ascent is 18 and its speed is 250 feet per second. (a) Find the plane's altitude after 1 minute. (Round your answer to the nearest whole number.) 4635 ft (b) How long will it take the plane to climb to an altitude of 10,000 feet? (Round your answer to one decimal place.) sec 32 of 35 10/20/ :54 AM
33 62. Question Details LarTrig [ ] An airplane flying at 500 miles per hour has a bearing of 52. After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure? (Round your answers to the nearest whole number.) 462 miles north 591 miles east 63. Question Details LarTrig [ ] A jet leaves Reno, Nevada and is headed toward Miami, Florida at a bearing of 100. The distance between the two cities is approximately 2472 miles. (a) How far north and how far west is Reno relative to Miami? (Round your answers to two decimal places.) miles north miles west (b) If the jet is to return directly to Reno from Miami, at what bearing should it travel? Question Details LarTrig [ ] A surveyor wants to find the distance across a pond (see figure). The bearing from A to B is N 33 W. The surveyor walks x = 50 meters from A to C, and at the point C the bearing to B is N 68 W. (a) Find the bearing from A to C. N 57 E (b) Find the distance from A to B. (Round your answer to two decimal places.) m 65. Question Details LarTrig [ ] Find a model for simple harmonic motion satisfying the specified conditions. Displacement, d (t = 0) Amplitude, a Period 0 8 centimeters 2 seconds d = 33 of 35 10/20/ :54 AM
34 66. Question Details LarTrig [ ] A point on the end of a tuning fork moves in simple harmonic motion described by tuning fork for a certain note has a frequency of 248 vibrations per second. ω = d = a sin ωt. Find ω given that the 67. Question Details LarTrig [ ] A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of y = 8.5 feet from its low point to its high point (see figure), and that it returns to its high point every 8 seconds. Write an equation that describes the motion of the buoy if its high point is at t = 0. d = Solution or Explanation At t = 0, buoy is at its high point d = a cos(ωt). Distance from high to low = 2 a = 8.5 a = 17 4 Returns to high point every 8 seconds: 2π Period: = 8 ω π = ω 4 17 πt d = cos Question Details LarTrig [ ] For the simple harmonic motion described by the trigonometric function, find the maximum displacement, the frequency, the value of d when t = 5, and the least positive value of t for which d = 0. Use a graphing utility to verify your results. d = 8 cos 6π t 5 (a) Find the maximum displacement. 8 (b) Find the frequency. 3/5 cycles per unit of time (c) Find the value of d when t = 5. d = 8 (d) Find the least positive value of t for which d = 0. t = 5/12 34 of 35 10/20/ :54 AM
35 69. Question Details LarTrig [ ] For the simple harmonic motion described by the trigonometric function, find the maximum displacement, the frequency, the value of d when t = 4, and the least positive value of t for which d = 0. Use a graphing utility to verify your results. d = 1 3 sin 2πt (a) Find the maximum displacement. 1/3 (b) Find the frequency. 1 cycles per unit of time (c) Find the value of d when t = 4. d = 0 (d) Find the least positive value of t for which d = 0. t = 1/2 Assignment Details Name (AID): Practice Test 2 Part A Chap 1 Sections 5,6,7,8 ( ) Feedback Settings Submissions Allowed: 5 Before due date Category: Homework Question Score Code: Assignment Score Locked: No Publish Essay Scores Author: Smithies, Laura ( lsmithie@kent.edu ) Question Part Score Last Saved: Oct 19, :23 PM EDT Mark Permission: Protected Add Practice Button Randomization: Person Help/Hints Which graded: Last Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response 35 of 35 10/20/ :54 AM
Practice Test Part B Chapter Sections,,3 (5477) Question 3456789034567890345678930333334353 Description This is the second of two practice tests for tests two. The other is Part A and covers.5,.6,.7 and.8.
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