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 | 1. A student graphs f (x) = cos x on a graphing calculator and gets what appears to be a straight line at height y = 1 instead of the usual cosine curve. Upon investigation, you discover that the calculator has graphing window - 10 x 10, - 10 y 10 and is in degrees mode. Explain what went wrong and how to correct it. |
| | 2. Many students are comfortable using degrees to measure angles and don't understand why they must learn radian measures. As discussed in the text,
radians directly measure distance along the unit circle. Distance is an important aspect of many applications. In addition, we will see later that many calculus formulas are simpler in
radians form than in degrees. Aside from familiarity, discuss any and all advantages of degrees over
radians. On balance, which is better? |
| | 3. In
Example 5.7, we saw that the sum of a certain pair of
periodic functions turned out to be a periodic function, while the sum of a different pair of
periodic functions was not periodic. Explain why the change in coefficients from 3 to  made such a big difference. |
| | 4. The trigonometric functions can be defined in terms of the unit circle (as done in the text) or in terms of right triangles for angles between 0 and 
radians. In calculus and most scientific applications, the trigonometric functions are used to model
periodic phenomena (quantities that repeat). Given that we want to emphasize the
periodic nature of the functions, explain why we would prefer the circular definitions to the triangular definitions. |
| In exercises 5 and 6, convert the given
radians measure to degrees.
|
| 5. (a) /4 (b) /3 (c) /6 (d) 4 /3 | 6. (a) 3 /5 (b) /7 (c) 2 (d) 3 |
| In exercises 7 and 8, convert the given degrees measure to
radians.
|
| 7. (a) 180° (b) 270° (c) 120° (d) 30° | 8. (a) 40° (b) 80° (c) 450° (d) 390° |
| In exercises 9-26, sketch a graph of the
function.
|
| 9. f (x) = sin 2x | 10. f (x) = cos 3x |
| 11. f (x) = tan 2x | 12. f (x) = sec 3x |
| 13. f (x) = 3 cos (x - /2) | 14. f (x) = 4 cos (x + ) |
| 15. f (x) = xsin x | 16. f (x) = sin x2 |
| 17. f (x) = sin 2x - 2 cos 2x | 18. f (x) = cos 3x - sin 3x |
| 19. f (x) = cos 5x + 3 sin 5x | 20. f (x) = 4 cos x - sin x |
| 21. f (x) = cos x -  cos 8x | 22. f (x) = sin 2x + sin 16x |
| 23. f (x) = sin xsin 12x | 24. f (x) = sin xcos 12x |
| 25. f (x) = cos 2x + sin x | 26. f (x) = 2 sin 3x - cos x
|
| In exercises 27-30, identify the
domain and range of the function.
|
| 27. f (x) = tan x | 28. f (x) = cot x |
| 29. f (x) = sec x | 30. f (x) = csc x
|
| In exercises 31-38, identify the
amplitude, period, and frequency.
|
| 31. f (x) = 3 sin 2x | 32. f (x) = 2 cos 3x |
| 33. f (x) = 5 cos 3x | 34. f (x) = 3 sin 5x |
| 35. f (x) = 3 cos (2x - /2) | 36. f (x) = 4 sin (3x + ) |
| 37. f (x) = - 4 sin x | 38. f (x) = - 2 cos 3x
|
| In exercises 39-42, prove that the given trigonometric identity is true.
|
| 39. sin ( -  ) = sin cos - sin cos | 40. cos ( - ) = cos cos + sin sin |
| 41. sec 2 = tan 2 + 1 | 42. csc 2 = cot 2 + 1
|
| 43. Prove that, for some
constant , 4 cos x - 3 sin x = 5 cos (x + ). Then, estimate the value of .
|
| 44. Prove that, for some
constant ,
Then, estimate the value of .
|
| In exercises 45-48, determine whether or not the
function is periodic. If it is periodic, find the smallest (fundamental) period.
|
| 45. f (x) = cos 2x + 3 sin x | 46. f (x) = sin x - cos  x |
| 47. f (x) = sin 2x - cos 5x | 48. f (x) = cos 3x - sin 7x
|
| In exercises 49-56, find all solutions of the given equation.
|
| 49. 2 cos x -1 = 0 | 50. 2 sin x + 1 = 0 |
| 51. cos2 x + cos x = 0 | 52. sin2 x - sin x = 0 |
| 53. sin2 x - 4 sin x + 3 = 0 | 54. sin2 x - 2 sin x -3 = 0 |
| 55. sin2 x + cos x -1 = 0 | 56. sin 2 x - cos x = 0
|
| In exercises 57-60, use the
range for to determine the indicated
function value.
|
| 57.  , find cos . | 58.  find sin.
|
| 59.  find cos . | 60.  find tan.
|
| In exercises 61-68, use a graphing calculator or computer to determine the number of solutions of each equation, and numerically estimate the solutions (x is in radians).
|
| 61. 2 cos x = 2 - x | 62. 3 sin x = x |
| 63. cos x = x2 - 2 | 64. sin x = x2 |
| 65. sin x - x3 + 1 = 0 | 66. x4 - cos x = 0 |
| 67. x3 - xsin x = 0 | 68. x2 - xcos x = 0
|
| 69. A person sitting 2 miles from a rocket launch site measures 20° up to the current location of the rocket. How high up is the rocket? |
| 70. A person who is 6 feet tall stands 4 feet from the base of a light pole and casts a 2 - foot-long shadow. How tall is the light pole? |
| 71. A surveyor stands 80 feet from the base of a building and measures an angle of 50° to the top of the steeple on top of the building. The surveyor figures that the center of the steeple lies 20 feet inside the front of the structure. Find the height of the steeple. |
| 72. Suppose that the surveyor of exercise 53 estimates that the center of the steeple lies between 20' and 21' inside the front of the structure. Determine how much the extra foot would change the calculation of the height of the building. |
| 73. In an AC circuit, the voltage is given by v(t ) = vpsin 2 ft, where vp is the peak voltage and f is the
frequency in Hz. A voltmeter actually measures an average (called the root-mean-square) voltage, equal to vp/. If the voltage has
amplitude 170 and period /30, find the
frequency and meter voltage. |
| 74. An old-fashioned LP record player rotated records at 33 rpm (revolutions per minute). What is the period (in minutes) of the rotation? What is the period for a 45 rpm single? |
| 75. Suppose that the ticket sales of an airline (in thousands of dollars) is given by  where t is measured in months. What real-world phenomenon might cause the fluctuation in ticket sales modeled by the sine term? Based on your answer, what month corresponds to t = 0 ? Disregarding seasonal fluctuations, by what amount is the airline's sales increasing annually? |
| 76. Piano tuners sometimes start by striking a tuning fork and then the corresponding piano key. If the tuning fork and piano note each have
frequency 8, then the resulting sound is sin 8t + sin 8t . Graph this. If the piano is slightly out-of-tune at
frequency 8.1, the resulting sound is sin 8t + sin 8.1t . Graph this and explain how the piano tuner can hear the small difference in
frequency. |
| 77. Many graphing calculators and computers will “graph” inequalities by shading in all points (x, y) for which the inequality is true. If you have access to this capability, graph the inequality sin x < cos y.
|
| 78. Calculator and computer graphics can be inaccurate. Using an initial graphing window of - 1 x 1 and - 1 y 1, graph  . Describe the behavior of the graph |
| near x = 0. Zoom in closer and closer to x = 0, using a window with -0.001 x 0.001, then -0.0001 x 0.0001, then -0.00001 x 0.00001 and so on, until the behavior near x = 0 appears to be different. We don't want to leave you hanging: the initial graph gives you good information and the tightly zoomed graphs are inaccurate due to the computer's inability to compute tan x exactly. |
| 79. This exercise deals with parametric equations, which most graphing calculators will graph. Try graphing x(t ) = cos t and y(t ) = sin t . The graph will probably look like an ellipse. Use the definition of sine and cosine to explain why the graph is actually a circle. Adjust your graphing window so that the graph looks like a circle. (Hint: Adjust the window so that distances of 1 in the x - and y - directions are equivalent; that is, make the aspect ratio equal to 1.) |
 | 80. As seen in exercise 78, computer graphics can be misleading. This exercise works best using a “disconnected” graph (individual dots, not connected). Graph y = sin x2 using a graphing window for which each pixel represents a step of 0.1 in the x - or y - direction. You should get the impression of a sine wave that oscillates more and more rapidly as you move to the left and right. Next, change the graphing window so that the middle of the original screen (probably x = 0 ) is at the far left of the new screen. You will probably see what appears to be a
random jumble of dots. Continue to change the graphing window by increasing the x - values. Describe the patterns or lack of patterns that you see. You should find one pattern that looks like two rows of dots across the top and bottom of the screen; another pattern looks like the original sine wave. For each pattern that you find, pick adjacent points with x - coordinates a and b. Then change the graphing window so that a x b and find the portion of the graph that is missing. Remember that, whether the points are connected or not, computer graphs always leave out part of the graph; it is part of your job to know whether the missing part is important or not. |
| | 81. Compare and contrast the graphs of sin x + sin 2 x and sin 3x + sin 2 x. Based on your results (don't draw any more graphs!) predict what the graph of sin x + sin 2 x will look like on the interval [100,102]. Can you accurately predict the shape of sin 3x + sin 2 x on the same interval? |
| | 82. In his book and video series The Ring of Truth, physicist Philip Morrison performed an experiment to estimate the circumference of the earth. In Nebraska, he measured the angle to a bright star in the sky, drove 370 miles due south into Kansas, and measured the new angle to the star. Some geometry shows that the difference in angles, about 5.02°, equals the angle from the center of the earth to the two locations in Nebraska and Kansas. If the earth is perfectly circular (it's not) and the circumference of the portion of the circle measured out by 5.02° is 370 miles, estimate the circumference of the earth. This experiment was based on a similar experiment by the ancient Greek scientist Eratosthenes. The ancient Greeks and the Spaniards of Columbus's day knew that the earth was round, they just disagreed about the circumference. Columbus argued for a figure about half of the actual value, since a ship couldn't survive on the water long enough to navigate the true
distance. |
