11.5: Radians
( \newcommand{\kernel}{\mathrm{null}\,}\)
6.1 Arclength and Radians
Homework 6.1
1.
Radians | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3\pi}{4} | \pi | \dfrac{5\pi}{4} | \dfrac{3\pi}{2} | \dfrac{7\pi}{4} | 2\pi |
Degrees | 0^{\circ} | 45^{\circ} | 90^{\circ} | 135^{\circ} | 180^{\circ} | 225^{\circ} | 270^{\circ} | 315^{\circ} | 360^{\circ} |
3.
a 120^{\circ}=\dfrac{2 \pi}{3} radians
b 240^{\circ}=\dfrac{4 \pi}{3} radians
c 480^{\circ}=\dfrac{8 \pi}{3} radians
d 600^{\circ}=\dfrac{10 \pi}{3} radians
5.
a 45^{\circ}=\dfrac{\pi}{4} radians
b 135^{\circ}=\dfrac{3 \pi}{4} radians
c 225^{\circ}=\dfrac{5 \pi}{4} radians
d 315^{\circ}=\dfrac{7 \pi}{4} radians
7.
9.
a 0.52
b 2.62
c 3.67
d 5.76
11.
13. 2.09
15. 2.62
17. 0.52
19. 2.36
21.
a II
b IV
c IV
d I
23.
a III
b II
c I
d IV
25.
Radians | \dfrac{\pi}{6} | \dfrac{\pi}{4} | \dfrac{\pi}{3} |
Degrees | 30^{\circ} | 45^{\circ} | 60^{\circ} |
27.
Radians | \dfrac{7\pi}{6} | \dfrac{5\pi}{4} | \dfrac{4\pi}{3} |
Degrees | 210^{\circ} | 225^{\circ} | 240^{\circ} |
29.
a 1.31
b 4.12
c 5.71
31.
a 45.8^{\circ}
b 200.5^{\circ}
c 292.2^{\circ}
33. 5.86 in
35. 4.13 m
37. 160.42^{\circ}
39.
a \dfrac{5\pi}{6}
b 32.72 ft
41. \dfrac{8}{67} radians \approx 6.84^{\circ}
43.
a 33,000 \pi \approx 103,672.6 in
b 33,000 \pi \approx 103.672 .6 in per min
45. 170 \pi \approx 534.1 \mathrm{~m} per min
47.
(0.2,0.98),(0.2,-0.98)
49.
(0.94,-0.35),(-0.94,-0.35)
51.
\left(\dfrac{-\sqrt{3}}{2}, \dfrac{1}{2}\right),\left(\dfrac{-\sqrt{3}}{2}, \dfrac{-1}{2}\right)
53.
a
b
\theta | 1 | 2 | 3 | 4 | 5 | 6 |
s | 4 | 8 | 12 | 16 | 20 | 24 |
c
m = 4
d Arclength doubles; arclength triples
55.
a \dfrac{\pi}{10} radians per min
b \dfrac{10 \pi}{9} radians per sec
57.
a \dfrac{\theta}{2 \pi}
b \dfrac{3}{8}, \dfrac{5}{6}, \dfrac{7}{12}
59. 32.5 \mathrm{~cm}^2
6.2 The Circular Functions
Homework 6.2
1.
a | b | c | d | |
t | \dfrac{\pi}{4} | \dfrac{3\pi}{4} | \dfrac{5\pi}{4} | \dfrac{7\pi}{4} |
x | \dfrac{1}{\sqrt{2}} | \dfrac{-1}{\sqrt{2}} | \dfrac{-1}{\sqrt{2}} | \dfrac{1}{\sqrt{2}} |
y | \dfrac{1}{\sqrt{2}} | \dfrac{1}{\sqrt{2}} | \dfrac{-1}{\sqrt{2}} | \dfrac{-1}{\sqrt{2}} |
3.
a | b | c | d | |
t | \dfrac{\pi}{3} | \dfrac{2\pi}{3} | \dfrac{4\pi}{3} | \dfrac{5\pi}{3} |
x | \dfrac{1}{2} | \dfrac{-1}{2} | \dfrac{-1}{2} | \dfrac{1}{2} |
y | \dfrac{\sqrt{3}}{2} | \dfrac{\sqrt{3}}{2} | \dfrac{-\sqrt{3}}{2} | \dfrac{-\sqrt{3}}{2} |
5.
a \sin 0.4 \approx 0.39, \cos 0.4 \approx 0.92, \tan 0.4 \approx 0.42
b \sin 1.2 \approx 0.93, \cos 1.2 \approx 0.36, \tan 1.2 \approx 2.6
c \sin 2 \approx 0.91, \cos 2 \approx -0.42, \tan 2 \approx -2.2
7.
a \sin 2.8 \approx 0.33, \cos 2.8 \approx -0.94, \tan 2.8 \approx -0.36
b \sin 3.5 \approx -0.35, \cos 3.5 \approx -0.94, \tan 3.5 \approx 0.37
c \sin 5 \approx -0.96, \cos 5 \approx 0.28, \tan 5 \approx -3.3
9. t \approx 1.27 or t \approx 5
11. t \approx 3.92 or t \approx 5.5
13. t \approx 2.72 or t \approx 5.87
15. II
17. II
19. III
21. Negative
23. Positive
25. Positive
27. \sin 3.5, \sin 0.5, \sin 2.5, \sin 1.5
29. \cos 3, \cos 4, \cos 2, \cos 5
31. January 1: 4.24, April 1: 6.45, July 1: 8:02, October 1: 5:55
33. 1.34
35. 0.84
37. 0.02
39. \dfrac{1}{12}\pi
41. \dfrac{1}{3}\pi
43. \dfrac{1}{4}\pi
45.
a \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}
b \(\dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4\)
c \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}
47.
\theta | \sin \theta | \cos \theta | \tan \theta |
\dfrac{7\pi}{6} | \dfrac{-1}{2} | \dfrac{-\sqrt{3}}{2} | \dfrac{1}{\sqrt{3}} |
\dfrac{5\pi}{4} | \dfrac{-1}{\sqrt{2}} | \dfrac{-1}{\sqrt{2}} | 1 |
\dfrac{4\pi}{3} | \dfrac{-\sqrt{3}}{2} | \dfrac{-1}{2} | \sqrt{3} |
49. \dfrac{1}{4}
51. - \dfrac{3 + \sqrt{3}}{3}
53. \dfrac{3 - 6 \sqrt{3}}{4}
55. (\cos 2.5, \sin 2.5) \approx (-0.8, 0.6)
57. (\cos 8.5, \sin 8.5) \approx (-0.6, 0.8)
59. \cos 5 \approx 0.28 mi east, \sin 5 \approx -0.96 mi north, or about 0.96 mi south
61. 1.75
63. 5.8
65. 3.84
67.
a
Intersection: (\dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}) and (\dfrac{-1}{\sqrt{2}}, \dfrac{-1}{\sqrt{2}})
b (\cos \dfrac{\pi}{4}, \sin \dfrac{\pi}{4}) and (\cos \dfrac{5\pi}{4}, \sin \dfrac{5\pi}{4})
69.
a
m = \dfrac{3}{8}
b \tan ^{-1}\left(\dfrac{3}{8}\right) \approx 0.3588
71. y-2=\sqrt{3}(x-4)
73. y+8=(\tan 2.4)((x-5) or y+8=-0.916(x-5)
75. Any point (x, y) on the terminal side of \theta satisfies \cos \theta=\frac{x}{r}, \sin \theta=\frac{y}{r}. For the point P where r=1, \cos \theta=x, \sin \theta=y. The arc of length t is spanned by an angle \theta in standard position. Because arclength is r \theta and r=1, t=\theta, so x=\cos t, y=\sin t.
77. The two right triangles shown are similar, so their sides are proportional. The hypotenuse of the large triangle is r times the hypotenuse of the small triangle, so the two legs of the large triangle must be r times the legs of the small triangle. Thus, because the coordinates of the vertex on the unit circle are (\cos \theta, \sin \theta), the coordinates of P must be (r \cos \theta, r \sin \theta).
79. 71 \mathrm{~m} west, 587 \mathrm{~m} north
6.3 Graphs of the Circular Functions
Homework 6.3
1.
a
\theta | 0 | \dfrac{\pi}{12} | \dfrac{\pi}{6} | \dfrac{\pi}{4} | \dfrac{\pi}{3} | \dfrac{5\pi}{12} | \dfrac{\pi}{2} | \dfrac{7\pi}{12} | \dfrac{2\pi}{3} | \dfrac{3\pi}{4} | \dfrac{5\pi}{6} | \dfrac{11\pi}{12} | \pi |
\cos \theta | 1 | 0.97 | 0.87 | 0.71 | 0.50 | 0.26 | 0 | -0.26 | -0.50 | -0.71 | -0.87 | -0.97 | -1 |
b
3.
5.
a
b Domain: (-\infty, \infty), range: [-1,1]
7.
a
b Domain: x \neq \dfrac{n \pi}{2}, n an odd integer, range: (-\infty, \infty)
9.
a x \approx 0.7 or x \approx 2.4
b x \approx 0.36 or x \approx 2.78
11.
a x \approx 2 or x \approx 4.3
b x \approx 2.5 or x \approx 3.79
13. x \approx 1.3 or x \approx 4.5
15. x \approx 2.7 or x \approx 5.8
17. x \approx 1.4 or x \approx 4.5
19. x \approx 2.2 or x \approx 5.3
21. I: 0.5, II: 2.7, III: 3.6, IV: 5.8
23. I: 0.6, II: 2.6, III: 3.7, IV: 5.7
25. I: 1.3, II: 1.8, III: 4.5, IV: 4.9
27. t \approx 0.74 or t \approx 5.55
29. t \approx 1.01 or t \approx 4.15
31. x \approx 3.94 or x \approx 5.48
33. t=\dfrac{3 \pi}{2}
35. x=\dfrac{\pi}{4} or x=\dfrac{5 \pi}{4}
37. z=\dfrac{\pi}{3} or z=\dfrac{5 \pi}{3}
39. s=\dfrac{2 \pi}{3} or s=\dfrac{5 \pi}{3}
41. t=\dfrac{5 \pi}{4} or t=\dfrac{7 \pi}{4}
43. x=\dfrac{5 \pi}{6} or x=\dfrac{7 \pi}{6}
45. a 0.78 b 1.12
47. a 0.26 b 1.28
49. a −0.9 b No solution
51.
a \dfrac{1}{\sqrt{2}}
b 0.9
53. -6 \sqrt{2}
55. -4 \sqrt{3}
57. 6
59.
b-c.
d. t \approx 10 and t \approx 20
e. t \approx 7.5 to t \approx 22
61.
b-c.
d. High: day 204,105^{\circ}; low: day 25,66^{\circ}
e. d \approx 128 to d \approx 281
63.
a -0.8,0.6, \dfrac{-4}{3}
b 0.8,-0.6, \dfrac{-4}{3}
c -0.8,-0.6, \dfrac{4}{3}
65.
a 0.92,-0.39, \dfrac{-92}{39}
b -0.92,0.39, \dfrac{-92}{39}
c 0.92,0.39, \dfrac{92}{39}
67.
69.
71.
a
b Domain: (-\infty, \infty), range: (-\infty, 9]
73.
a
b Domain: x \neq 0, range: (-\infty, 2)
75.
a
b Domain: [6, \infty), range: [0, \infty)
77.
a
b Domain: [-2,2], range: [-2,0]
79.
a
x | 0 | \dfrac{\pi}{2} | \pi | \dfrac{3\pi}{2} | 2\pi |
\cos x | 1 | 0 | -1 | 0 | 1 |
b Domain: (-\infty, \infty), Rangee: [-1,1]
6.4 Chapter 6 Summary and Review
Chapter 6 Review Problems
1.
a \dfrac{5 \pi}{12}
b \dfrac{7 \pi}{6}
c \dfrac{17 \pi}{9}
3.
a 0.47
b 2.48
c 3.80
5.
a 150^{\circ}
b 54^{\circ}
c 230^{\circ}
7.
a 114.59^{\circ}
b 206.26^{\circ}
c 45.84^{\circ}
9.
a \dfrac{4 \pi}{3}
b \dfrac{7\pi}{6}
c \dfrac{9 \pi}{4}
11.
a \dfrac{1}{8}
b \dfrac{5}{16}
c \dfrac{7}{6}
13.
a II
b I
c IV
15.
a 0.006, 2.17, 0.0379
b 0.0379
17. 6885 mph
19.
a 0
b \dfrac{-8}{\sqrt{3}}
c \dfrac{-1}{2}
21.
a (0.5, 0.8)
b (−0.4, 0.9)
c (−1.0, 0.1)
23.
a (r \cos \alpha, r \sin \alpha)
b (-r \cos \alpha, r \sin \alpha)
c (-r \cos \alpha,-r \sin \alpha)
d (r \cos \alpha,-r \sin \alpha)
25. 6 \pi
27. >
29. <
31. 9.86
33. −1.33
35.
a \dfrac{\pi}{6}
b \dfrac{\pi}{4}
c \dfrac{3 \pi}{8}
d \dfrac{5 \pi}{12}
37.
a 0.34
b 0.76
c 1.25
d 1.5
39.158.2^{\circ}
41.
43.
a mid: y=5, amp: 3, period: \pi
b 0.86, 2.28, 4.00, 5.42
45.
a mid: y=10, amp: 4.8, period: 2\pi
b 1.93, 4.2
47. \dfrac{5 \pi}{12}, \dfrac{17 \pi}{12}
49. \dfrac{\pi}{3}, \dfrac{2 \pi}{3}
51. \pi
53. 1.37, 4.51
55. 6.02, 3.40
57. 0.32, 5.97
59.
a 1.21, 5.07
b 0.9394
61.
a 0.40, 2.74
b 0.3827
63. Dom: all real numbers, Rge: y ≥ 4
65. Dom: −4 ≤ s ≤ 4, Rge: −4 ≤ y ≤ 0
67.
a x^2+y^2=1
b (\cos t, \sin t)
c \cos ^2 t+\sin ^2 t=1
d Yes