Note: These problems are in fact Chapter 6 problems starting with the 8th edition. The author has renumbered some of the early chapters.
(sections numbers are 6.1 - 6.6 starting with the 8th edition)
Note: Throughout this page `theta stands for the lower-case Greek letter theta (q), `phi for the lower-case Greek letter phi (f), `omega for the lower-case Greek letter omega (w), `pi for the lower-case Greek letter `pi (p). These symbols are represented in this manner because your browser might not represent them correctly.
Sec. 5.1 (6.1 starting with 8th edition)
Draw each angle:
6. 540 deg
12. 21 `pi / 4
Convert the angle from degrees to radians or from radians to degrees:
18. -30 deg
24. -180 deg
30. 4 `pi
36. -3 `pi / 4
48. 51 deg
50. 200 deg
54. `pi
If r is the radius of a circle and s the length of the arc subtended by the angle `theta, then find the missing quantity:
40. `theta = 1 / 4 radian, s = 6 cm, r = ?
42. r = 6 meters, s = 8 meters, `theta = ?
78. The radius of each wheel of a car is 15 inches. If the wheels are turning at 3 revolutions / second, how fast is the car moving? Express your answer in inches per second and in miles per hour.
80. The windshield wiper of a car is 18 inches long. If it takes 1 second to trace out 1 / 3 revolution, how fast is the tip of the wiper moving?
84. A Ferris wheel has radius 30 feet and makes one revolution in 70 seconds. What is the speed of a point on the outer part of the wheel?
90. A nautical mile is the length of the arc subtended by a central angle of 1 minute on a great circle (a circle on the surface of the Earth which divides the Earth into two equal hemispheres). If the radius of the Earth is 3960 miles, express 1 nautical mile in terms of statute miles (a statue mile is 5280 feet).
Section 5.2 (6.2 starting with 8th edition)
For the given points on the terminal side of angle `theta, find the exact value of the six trigonometric functions of `theta:
6. ( 1, -1)
10. (-0.3, -0.4)
Without using a calculator find the exact value of each expression:
12. sin 30 deg - cos 45 deg
18. sec (30 deg) * cot (45 deg)
20. 5 cos( 90 deg) - 8 sin( 270 deg)
24. tan(`pi/3) + cos(`pi/3)
30. sec `pi - csc (`pi/2)
Find the exact values of the six trigonometric functions of:
36. - `pi / 3
40. 3 `pi
42. -270 deg
If `theta = 60 deg find the exact value of:
72. cos(`theta)
78. cos( 2 * `theta)
80. 2 cos(`theta)
84. Find the exact value of tan 60 deg + tan 150 deg.
90. If cos( `theta) = 2/3, find sec (`theta).
Sec 5.3 (6.3 starting with 8th edition)
Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator.
6. Sec 540`theta deg
10. sin 9 `pi / 4
12. csc 9 `pi /2
name the quadrant in which the angle `theta lies.
18. sin `theta < 0, cos `theta > 0
20. cos `theta > 0, tan `theta > 0
24. csc `theta > 0, cos `theta < 0
in the next problem sin `theta and cos `theta are given. Find the exact value of each of the four remaining trigonometric functions.
30. sin `theta = 2`sqrt(2) / 3 , cos `theta = -1/3
find the exact value of each of the remaining trigonometric functions of `theta.
36. sin `theta = -5/13, `theta in quadrant 3
40. sin `theta = -2/3, `pi < `theta < 3`pi/2
42. cos `theta = -1/4, tan `theta > 0
48. sec `theta = -2, tan `theta > 0
use the even- odd properties to find the exact value of each expression. Do not use a calculator.
50. cos (-30 deg )
54. csc (-30 deg)
60. sin (-`pi/3)
66. csc (-`pi/3)
find the exact value of each expression. Do not use a caculator.
70. tan (-6`pi) + cos (9`pi/4)
72. cos (-17`pi/4) - sin (-3 `pi/2)
78. cot 20 deg - cos 20 deg/sin20 deg
80. If cos `theta = 0.2, find the value of cos `theta + cos ( `theta+2`pi) + cos (`theta+4`pi).
84. What is the domain of the cosine function?
90. What is the range of the cosine function?
96. If the cosine function even, odd or neither? Is its graph symmetric? With respect to what?
100. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?
102. If f(x) = cos x and f(a) = 1/4, find the exact values of:
110. Use the periodic and even- odd properties to show that the range of the cotangent function is the set of all real numbers.
Sections 5.4 and 5.5 (6.4 and 6.5 starting with 8th edition)
Refer to the graphs to answer each question, if necessary.
6. What is the smallest value of y = cos x?
10. for what numbers x, -2`pi <= x <= 2`pi, does cos x =1? What about cos x = -1?
12. What is the y- intercept of y = cot x?
18. for what numbers x, -2`pi <= x <= 2`pi, does the graph of y= csc x have vertical asymptotes?
20. For what numbers x, -2`pi <= x <= 2`pi,does the graph of y= cot x have vertical asymptotes?
Use transformations to graph each of the following functions:
36. y =Cos x + 1
40. y = cos`pi / 2x
42. y =3 cos x + 3
48. y = -cot x
50. y = csc (x - `pi)
54. y = 4tan 1/2 x
60. y = -3 tan 2x
63. On the same coordinate axes, graph y =2 sin x and y = sin 2x, 0 <= x <= 2 `pi.
Compare each graph’s maximum and minimum value.
Compare each graph’s period.
66. Repeat problem 63 for y = 4 cos x and y = cos 4x, 0 <= x <= 2`pi.
70. Graph y = 2 sin x, y = 1/2 sin x and y = 8 sin x. What do you conclude about the graph of y =A sin x, A > 0 ?
72. Graph y = sin x, y = sin [ x - ( `pi / 3 ) ], y = sin { x - ( `pi / 4 )}, and y = sin [ x - ( `pi / 6 ) ]. What d you conclude about the graph of y = sin ( x -`phi ), `phi > 0?
Section 5.6 (6.6 starting with 8th edition)
Determine the amplitude and period of each function without graphing.
6. y = -3 cos 3x
10. y = 9 / 5 cos ( - 3 `pi/ 2 x )
Match the given functions to one of the graphs (A) – (J).
12. y = 2 cos (`pi/ 2x)
18. y = - 2 cos( `pi /2 x)
20. y = -2 sin (1/2 x)
Match the given function to one of the graphs (A)-(F)
24. y =3 sin 2 x
Graph each function.
30. y = 2 sin `pi x
36. y =( 4/3) cos (-1 / 3 * x )
Find the amplitude, period, and phase shift of each function. Use transformation to graph the function. Show at least one period.
54. y = 3 cos (2x + `pi)
60. y = 2 cos (2`pi (x - 4 ) )
Write the equation of a sine function that has the given characteristics.
66. Amplitude: 4
Period: 1
70. Amplitude: 2
Period: `pi
Phase shift: -2
72. Alternating current (AC) circuits. The current I, in amperes, flowing through an ac (alternating current) circuit at time t is
What is the period? What is the amplitude? Graph the function over two periods.
80. Biorhythms. In the theory of biorhythms a sine function of the form P = 50 sin ( t ) + 50 is used to measure the percent P of a person's potential at time t, where t is measured in days starting with the person's day of birth. Three characteristics are commonly measured:
a) Find for each characteristic
b) Graph all three functions
c) Is there a time t when all three characteristics have 100% potential? When is this?
d) Suppose that you are 20 years old today. Describe your physical, emotional, and intellectual potential for the next 30 days.
80. Monthly temperature. The following data represent the average monthly temperatures for Washington, D.C.:
a) Draw a scatter diagram of the data for one period.
b) Find a sinusoidal function of the form y = A sin( `omega x - `phi ) + B that fits the data.
c) Draw the sinusoidal function found in (b) on the scatter diagram.
d) Use a graphing utility to find the sinusoidal function of best fit.
e) Draw the sinusoidal function of best fit on the scatter diagram.