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11.5: Radians

( \newcommand{\kernel}{\mathrm{null}\,}\)

6.1 Arclength and Radians

Homework 6.1

1.

Screen Shot 2023-02-15 at 12.57.43 AM.png

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. Screen Shot 2023-02-15 at 12.57.54 AM.png

9.

a 0.52

b 2.62

c 3.67

d 5.76

11. Screen Shot 2023-02-15 at 1.06.27 AM.png

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. Screen Shot 2023-02-16 at 12.33.05 AM.png

(0.2,0.98),(0.2,-0.98)

49. Screen Shot 2023-02-16 at 12.33.11 AM.png

(0.94,-0.35),(-0.94,-0.35)

51. Screen Shot 2023-02-16 at 12.33.17 AM.png

\left(\dfrac{-\sqrt{3}}{2}, \dfrac{1}{2}\right),\left(\dfrac{-\sqrt{3}}{2}, \dfrac{-1}{2}\right)

53.

a Screen Shot 2023-02-16 at 12.50.43 AM.png

b

\theta 1 2 3 4 5 6
s 4 8 12 16 20 24

c Screen Shot 2023-02-16 at 12.51.52 AM.png

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}

Screen Shot 2023-02-19 at 12.22.32 AM.png

b \(\dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4\)

Screen Shot 2023-02-19 at 12.22.43 AM.png

c \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}

Screen Shot 2023-02-19 at 12.22.50 AM.png

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 Screen Shot 2023-02-19 at 12.27.45 AM.png

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 Screen Shot 2023-02-19 at 12.42.53 AM.png

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 Screen Shot 2023-02-19 at 12.54.29 AM.png

3. Screen Shot 2023-02-19 at 12.57.54 AM.png

5.

a Screen Shot 2023-02-19 at 12.58.31 AM.png

b Domain: (-\infty, \infty), range: [-1,1]

7.

a Screen Shot 2023-02-19 at 12.58.37 AM.png

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. Screen Shot 2023-02-19 at 1.17.17 AM.png

d. t \approx 10 and t \approx 20

e. t \approx 7.5 to t \approx 22

61.

b-c. Screen Shot 2023-02-19 at 1.17.22 AM.png

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. Screen Shot 2023-02-19 at 1.21.46 AM.png

69. Screen Shot 2023-02-19 at 1.21.53 AM.png

71.

a Screen Shot 2023-02-19 at 1.22.06 AM.png

b Domain: (-\infty, \infty), range: (-\infty, 9]

73.

a Screen Shot 2023-02-19 at 1.22.16 AM.png

b Domain: x \neq 0, range: (-\infty, 2)

75.

a Screen Shot 2023-02-19 at 1.22.23 AM.png

b Domain: [6, \infty), range: [0, \infty)

77.

a Screen Shot 2023-02-19 at 1.22.32 AM.png

b Domain: [-2,2], range: [-2,0]

79.

a Screen Shot 2023-02-25 at 1.05.22 PM.png

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. Screen Shot 2023-02-25 at 1.20.31 PM.png

43.

a Screen Shot 2023-02-25 at 1.25.00 PM.png mid: y=5, amp: 3, period: \pi

b Screen Shot 2023-02-25 at 1.25.05 PM.png 0.86, 2.28, 4.00, 5.42

45.

a Screen Shot 2023-02-25 at 1.25.23 PM.png mid: y=10, amp: 4.8, period: 2\pi

b Screen Shot 2023-02-25 at 1.25.31 PM.png 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. Screen Shot 2023-02-25 at 1.31.06 PM.png Dom: all real numbers, Rge: y ≥ 4

65. Screen Shot 2023-02-25 at 1.31.16 PM.png 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


This page titled 11.5: Radians is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Katherine Yoshiwara via source content that was edited to the style and standards of the LibreTexts platform.

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