11.5: Radians
- Page ID
- 122927
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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