11.5: Radians

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