Chapter Review
437
Chapter Review Things to Know Definitions Angle in standard position (p. 356)
Vertex is at the origin; initial side is along the positive x-axis
1 Degree 11°2 (p. 357)
1° =
1 Radian (p. 360)
1 revolution 360
The measure of a central angle of a circle whose rays subtend an arc whose length is the radius of the circle P = 1x, y2 is the point on the unit circle corresponding to u = t radians.
Trigonometric functions (pp. 371–372)
sin t = sin u = y csc t = csc u = Trigonometric functions using a circle of radius r (pp. 382–383)
1 , y
cos t = cos u = x y Z 0
sec t = sec u =
1 , x Z 0 x
tan t = tan u =
y , x Z 0 x
cot t = cot u =
x , y
y Z 0
For an angle u in standard position P = 1x, y2 is the point on the terminal side of u that is
also on the circle x2 + y2 = r2. y r r csc u = , y Z 0 y sin u =
x r r sec u = , x Z 0 x cos u =
y ,x Z 0 x x cot u = , y Z 0 y tan u =
f1u + p2 = f1u2, for all u, p 7 0, where the smallest such p is the fundamental period
Periodic function (p. 391) Formulas 1 revolution = 360°
(p. 358)
= 2p radians (p. 361) s = r u (p. 360) A =
u is measured in radians; s is the length of arc subtended by the central angle u of the circle of radius r; A is the area of the sector.
1 2 r u (p. 364) 2
v = rv (p. 365)
v is the linear speed along the circle of radius r; v is the angular speed (measured in radians per unit time).
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CHAPTER 5
Trigonometric Functions
TABLE OF VALUES U (Radians)
U (Degrees)
sin U
cos U
tan U
csc U
0
Not defined
0
0°
0
1
p 6
30°
1 2
23 2
23 3
p 4
45°
22 2
22 2
p 3
60°
23 2
p 2
90°
p 3p 2
sec U
cot U
1
Not defined
2
2 23 3
23
1
22
22
1
1 2
23
2 23 3
2
23 3
1
0
Not defined
180°
0
-1
270°
-1
0
1
Not defined
0
Not defined
-1
Not defined
-1
Not defined
0 Not defined 0
Fundamental Identities (p. 394) tan u =
sin u cos u , cot u = cos u sin u
csc u =
1 1 1 , sec u = , cot u = sin u cos u tan u
sin2 u + cos2 u = 1, tan2 u + 1 = sec2 u, 1 + cot2 u = csc2 u Properties of the Trigonometric Functions y = sin x Domain: - q 6 x 6 q (p. 404) Range: -1 … y … 1 Periodic: period = 2p 1360°2 Odd function
y = cos x (p. 406)
y = tan x (p. 420)
y = cot x (p. 422)
Domain: - q 6 x 6 q Range: -1 … y … 1 Periodic: period = 2p 1360°2 Even function
p Domain: - q 6 x 6 q , except odd multiples of 190°2 2 Range: - q 6 y 6 q Periodic: period = p 1180°2 Odd function
Domain: - q 6 x 6 q , except integer multiples of p 1180°2 Range: - q 6 y 6 q Periodic: period = p 1180°2 Odd function
y 1
– 2
–
1
2
5–– 2
x
y 1
– 2
1
2
3–– 2
2
5–– 2
x
y 1 5–– 2
3–– 2
–
––
2 1
2
3–– 2
5–– 2
x
5 ––– 2
x
y 1 3 ––– 2
–
2 1
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– 2
3 ––– 2
Chapter Review
y = csc x (p. 422)
Domain: - q 6 x 6 q , except integer multiples of p 1180°2 Range: ƒ y ƒ Ú 1 Periodic: period = 2p 1360°2 Odd function
y 1 3 –––
p Domain: - q 6 x 6 q , except odd multiples of 190°2 2 Range: ƒ y ƒ Ú 1 Periodic: period = 2p 1360°2 Even function
–
1
2
y = sec x (p. 423)
439
x
2
y 1 3 ––– 2
–– 2
1
– 2
3–– 2
x
Sinusoidal graphs (pp. 409 and 426) 2p v
y = A sin1vx2, v 7 0
Period =
y = A cos1vx2, v 7 0
Amplitude = ƒ A ƒ f Phase shift = v
f y = A sin1vx - f2 = A sincva x - b d v y = A cos1vx - f2 = A cosc vax -
f bd v
Objectives Section 5.1
5.2
1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓
5.3
6 ✓ 7 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓
You should be able to
Á
Review Exercises
Convert between degrees, minutes, seconds, and decimal forms for angles (p. 358)
82
Find the arc length of a circle (p. 360)
83, 84
Convert from degrees to radians and from radians to degrees (p. 361)
1–8
Find the area of a sector of a circle (p. 364)
83
Find the linear speed of an object traveling in circular motion (p. 365)
85–88
Find the exact values of the trigonometric functions using a point on the unit circle (p. 372)
79
Find the exact values of the trigonometric functions of quadrantal angles (p. 374) p Find the exact values of the trigonometric functions of = 45° (p. 376) 4 p p = 60° (p. 377) Find the exact values of the trigonometric functions of = 30° and 6 3 p p Find the exact values of the trigonometric functions for integer multiples of = 30°, = 45°, 6 4 p and = 60° (p. 380) 3 Use a calculator to approximate the value of a trigonometric function (p. 381)
9–15
Use circle of radius r to evaluate the trigonometric functions (p. 388)
80
Determine the domain and the range of the trigonometric functions (p. 388)
81
Determine the period of the trigonometric functions (p. 390)
81
Determine the signs of the trigonometric functions in a given quadrant (p. 392)
77–78
Find the values of the trigonometric functions using fundamental identities (p. 393)
21–30
Find the exact values of the trigonometric functions of an angle given one of the functions and the quadrant of the angle (p. 395)
31–46
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17, 18, 20 9, 11, 13, 15, 16, 19
13–16, 94 75, 76
440
CHAPTER 5 6 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 1 ✓
5.4
5.5
2 ✓ 1 ✓
5.6
2 ✓
Trigonometric Functions
Use even–odd properties to find the exact values of the trigonometric functions (p. 398)
27–30
Graph transformations of the sine function (p. 403)
47, 50
Graph transformations of the cosine function (p. 405)
48, 49
Determine the amplitude and period of sinusoidal functions (p. 408)
59–64, 89
Graph sinusoidal functions using key points: (p. 410)
47, 48, 63-64, 89
Find an equation for a sinusoidal graph (p. 413)
71–74
Graph transformations of the tangent function and cotangent function (p. 419)
51–56
Graph transformations of the cosecant function and secant function (p. 422)
57, 58
Graph sinusoidal functions of the form y = A sin1vx - f2 using the amplitude, period and phase shift (p. 425)
65–70, 90
Find a sinusoidal function from data (p. 429)
91–93
Review Exercises In Problems 1–4, convert each angle in degrees to radians. Express your answer as a multiple of p. 1. 135°
2. 210°
3. 18°
4. 15°
In Problems 5–8, convert each angle in radians to degrees. 5.
3p 4
6.
2p 3
7. -
5p 2
8. -
3p 2
In Problems 9–30, find the exact value of each expression. Do not use a calculator. 9. tan
p p - sin 4 6
10. cos
12. 4 cos 60° + 3 tan 15. sec a 18. cos
p 3
p 5p b - cota b 3 4
p p - csca - b 2 2
21. sin2 20° +
1 sec2 20°
24. tan 10° cot 10° 27. sin1-40°2 csc 40°
p p + sin 3 2
11. 3 sin 45° - 4 tan
13. 6 cos
3p p + 2 tana - b 4 3
14. 3 sin
16. 4 csc
3p p - cota - b 4 4
17. tan p + sin p
19. cos 540° - tan1-45°2 22.
1 cos2 40°
1 -
5p 2p - 4 cos 3 2
20. sin 630° + cos1-180°2 23. sec 50° cos 50°
cot2 40°
25. sec2 20° - tan2 20° 28. tan1-20°2 cot 20°
p 6
26.
1 2
sec 40°
29. cos 410° sec1-50°2
1 +
2
csc 40° 30. cot 200° tan1-20°2
In Problems 31–46, find the exact value of each of the remaining trigonometric functions. 31. sin u =
4 , 5
34. cot u =
12 , 5
u is acute
cos u 6 0
32. cos u =
3 , 5
5 35. sec u = - , 4
u is acute
tan u 6 0
33. tan u =
12 , 5
5 36. csc u = - , 3
sin u 6 0
cot u 6 0
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Chapter Review
37. sin u =
12 , u in quadrant II 13
40. cos u =
12 , 13
3 38. cos u = - , 5
3p 6 u 6 2p 2
41. tan u =
3p 6 u 6 2p 2
43. sec u = 3, 46. tan u = -2,
1 , 3
2 42. tan u = - , 3
180° 6 u 6 270°
44. csc u = -4, p 6 u 6
5 , 13
39. sin u = -
u in quadrant III
3p 2
441
3p 6 u 6 2p 2 90° 6 u 6 180° p 6 u 6 p 2
45. cot u = -2,
3p 6 u 6 2p 2
In Problems 47–58, graph each function. Each graph should contain at least one period. 47. y = 2 sin14x2
48. y = -3 cos12x2
51. y = tan1x + p2
52. y = -tana x -
55. y = cotax +
p b 4
49. y = -2 cos ax + p b 2
56. y = -4 cot12x2
p b 2
50. y = 3 sin1x - p2
53. y = -2 tan13x2
54. y = 4 tan12x2
57. y = sec ax -
58. y = cscax +
p b 4
p b 4
In Problems 59–62, determine the amplitude and period of each function without graphing. 59. y = 4 cos x
60. y = sin12x2
61. y = -8 sina
p xb 2
62. y = -2 cos13px2
In Problems 63–70, find the amplitude, period, and phase shift of each function. Graph each function. Show at least one period. 1 64. y = 2 cos a xb 3
63. y = 4 sin13x2 67. y =
1 3 sin a x - pb 2 2
68. y =
1 p 66. y = -cosa x + b 2 2
65. y = 2 sin12x - p2
3 cos16x + 3p2 2
69. y = -
2 cos1px - 62 3
70. y = -7 sina
4 p x + b 3 3
In Problems 71–74, find a function whose graph is given. 71.
72. y 5
73.
74. y 7
y 6
y 4
x
x 4
4
5
8
x
2
2
6
10
4
2 4 6
8 10
42
2
4 6 8 10 x
4 6
p 75. Use a calculator to approximate sin . Round the answer to 8 two decimal places. 76. Use a calculator to approximate sec 10°. Round the answer to two decimal places. 77. Determine the signs of the six trigonometric functions of an angle u whose terminal side is in quadrant III. 78. Name the quadrant u lies in if cos u 7 0 and tan u 6 0.
7
79. Find the exact values of the six trigonometric functions if 1 212 P = a- , b is the point on the unit circle that 3 3 corresponds to t. 80. Find the exact value of sin t, cos t, and tan t if P = (-2, 5) is the point on the circle that corresponds to t.
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442
CHAPTER 5
Trigonometric Functions
81. What is the domain and the range of the secant function? What is the period?
Month, m
Average Monthly Temperature, T
January, 1
51
February, 2
55
March, 3
63
83. Find the length of the arc subtended by a central angle of 30° on a circle of radius 2 feet. What is the area of the sector?
April, 4
67
May, 5
77
June, 6
86
84. The minute hand of a clock is 8 inches long. How far does the tip of the minute hand move in 30 minutes? How far does it move in 20 minutes?
July, 7
90
August, 8
90
September, 9
84
85. Angular Speed of a Race Car A race car is driven around a circular track at a constant speed of 180 miles 1 per hour. If the diameter of the track is mile, what is 2 the angular speed of the car? Express your answer in revolutions per hour (which is equivalent to laps per hour).
October, 10
71
November, 11
59
December, 12
52
82. (a) Convert the angle 32°20¿35– to a decimal in degrees. Round the answer to two decimal places. (b) Convert the angle 63.18° to D°M¿S– form. Express the answer to the nearest second.
86. Merry-Go-Rounds A neighborhood carnival has a merrygo-round whose radius is 25 feet. If the time for one revolution is 30 seconds, how fast is the merry-go-round going? 87. Lighthouse Beacons The Montauk Point Lighthouse on Long Island has dual beams (two light sources opposite each other). Ships at sea observe a blinking light every 5 seconds. What rotation speed is required to do this? 88. Spin Balancing Tires The radius of each wheel of a car is 16 inches. At how many revolutions per minute should a spin balancer be set to balance the tires at a speed of 90 miles per hour? Is the setting different for a wheel of radius 14 inches? If so, what is this setting? 89. Alternating Voltage The electromotive force E, in volts, in a certain ac (alternating circuit) circuit obeys the equation E = 120 sin1120pt2,
t Ú 0
where t is measured in seconds. (a) What is the maximum value of E? (b) What is the period? (c) Graph this function over two periods. 90. Alternating Current The current I, in amperes, flowing through an ac (alternating current) circuit at time t is I = 220 sina30pt + (a) (b) (c) (d)
p b, 6
t Ú 0
What is the period? What is the amplitude? What is the phase shift? Graph this function over two periods.
91. Monthly Temperature The following data represent the average monthly temperatures for Phoenix, Arizona.
SOURCE: U.S. National Oceanic and Atmospheric Administration
(a) Use a graphing utility to draw a scatter diagram of the data for one period. (b) By hand, find a sinusoidal function of the form y = A sin1vx - f2 + B that fits the data. (c) Draw the sinusoidal function found in part (b) on the scatter diagram. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on the scatter diagram. 92. Monthly Temperature The following data represent the average monthly temperatures for Chicago, Illinois.
Month, m
Average Monthly Temperature, T
January, 1
25
February, 2
28
March, 3
36
April, 4
48
May, 5
61
June, 6
72
July, 7
74
August, 8
75
September, 9
66
October, 10
55
November, 11
39
December, 12
28
SOURCE: U.S. National Oceanic and Atmospheric Administration
(a) Use a graphing utility to draw a scatter diagram of the data for one period. (b) By hand, find a sinusoidal function of the form y = A sin1vx - f2 + B that fits the data.
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Chapter Test
(c) Draw the sinusoidal function found in part (b) on the scatter diagram. (d) Use a graphing utility to find the sinusoidal function of best fit. (e) Graph the sinusoidal function of best fit on the scatter diagram. 93. Hours of Daylight According to the Old Farmer’s Almanac, in Las Vegas, Nevada, the number of hours of sunlight on the summer solstice is 13.367 and the number of hours of sunlight on the winter solstice is 9.667. (a) Find a sinusoidal function of the form
443
94. Unit Circle Fill in the angles (in degrees and radians) and terminal points P of each angle on the unit circle shown. y Angle:
Angle:
Angle: — 3
P
Angle:
P Angle:
Angle: — 4
P
Angle: — 6
P
P P
P
y = A sin1vx - f2 + B that fits the data. (b) Draw a graph of the function found in part (a). (c) Use the function found in part (a) to predict the number of hours of sunlight on April 1, the 91st day of the year. (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac and compare the actual hours of daylight to the results found in part (c).
P
Angle:
P
P
P Angle:
Angle: x
P
P P
Angle:
P P
Angle:
11 Angle: —— 6
Angle:
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7 Angle: —— 4 5 Angle: —— 3
