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Mathematics LibreTexts

5.2E: Exercises

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Verbal

1) Describe the unit circle.

Answer

The unit circle is a circle of radius 1 centered at the origin.

2) What do the x- and y-coordinates of the points on the unit circle represent?

3) Discuss the difference between a coterminal angle and a reference angle.

Answer

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, formed by the terminal side of the angle t and the horizontal axis.

4) Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

5) Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

Answer

The sine values are equal.

Algebraic

For the exercises 6-9, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by t lies.

6) sin(t)<0 and cos(t)<0

7) sin(t)>0 and cos(t)>0

Answer

I

8) sin(t)>0 and cos(t)<0

9) sin(t)<0 and cos(t)>0

Answer

IV

For the exercises 10-22, find the exact value of each trigonometric function.

10) sinπ2

11) sinπ3

Answer

32

12) cosπ2

13) cosπ3

Answer

12

14) sinπ4

15) cosπ4

Answer

22

16) sinπ6

17) sinπ

Answer

0

18) sin3π2

19) cosπ

Answer

1

20) cos0

21) cosπ6

Answer

32

22) sin0

Numeric

For the exercises 23-33, state the reference angle for the given angle.

23) 240°

Answer

60°

24) −170°

25) 100°

Answer

80°

26) −315°

27) 135°

Answer

45°

28) \dfrac{5π}{4}

29) \dfrac{2π}{3}

Answer

\dfrac{π}{3}

30) \dfrac{5π}{6}

31) −\dfrac{11π}{3}

Answer

\dfrac{π}{3}

32) \dfrac{−7π}{4}

33) \dfrac{−π}{8}

Answer

\dfrac{π}{8}

For exercises 34-49, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

34) 225°

Exercise \PageIndex{35}

300°

Answer

60°, Quadrant IV, \sin (300°)=−\dfrac{\sqrt{3}}{2}, \cos (300°)=\dfrac{1}{2}

36) 320°

37) 135°

Answer

45°, Quadrant II, \sin (135°)=\dfrac{\sqrt{2}}{2}, \cos (135°)=−\dfrac{\sqrt{2}}{2}

38) 210°

39) 120°

Answer

60°, Quadrant II, \sin (120°)=\dfrac{\sqrt{3}}{2}, \cos (120°)=−\dfrac{1}{2}

40) 250°

41) 150°

Answer

30°, Quadrant II, \sin (150°)=\frac{1}{2}, \cos(150°)=−\dfrac{\sqrt{3}}{2}

42) \dfrac{5π}{4}

Exercise \PageIndex{43}

\dfrac{7π}{6}

Answer

\dfrac{π}{6}, Quadrant III, \sin \left( \dfrac{7π}{6}\right )=−\dfrac{1}{2}, \cos \left (\dfrac{7π}{6} \right)=−\dfrac{\sqrt{3}}{2}

44) \dfrac{5π}{3}

45) \dfrac{3π}{4}

Answer

\dfrac{π}{4}, Quadrant II, \sin \left(\dfrac{3π}{4}\right)=\dfrac{\sqrt{2}}{2}, \cos\left(\dfrac{4π}{3}\right)=−\dfrac{\sqrt{2}}{2}

46) \dfrac{4π}{3}

47) \dfrac{2π}{3}

Answer

\dfrac{π}{3}, Quadrant II, \sin \left(\dfrac{2π}{3}\right)=\dfrac{\sqrt{3}}{2}, \cos \left(\dfrac{2π}{3}\right)=−\dfrac{1}{2}

48) \dfrac{5π}{6}

49) \dfrac{7π}{4}

Answer

\dfrac{π}{4}, Quadrant IV, \sin \left(\dfrac{7π}{4}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(\dfrac{7π}{4}\right)=\dfrac{\sqrt{2}}{2}

For the exercises 50-59, find the requested value.

50) If \cos (t)=\dfrac{1}{7} and t is in the 4^{th} quadrant, find \sin (t).

51) If \cos (t)=\dfrac{2}{9} and t is in the 1^{st} quadrant, find \sin (t).

Answer

\dfrac{\sqrt{77}}{9}

52) If \sin (t)=\dfrac{3}{8} and t is in the 2^{nd} quadrant, find \cos (t).

Exercise \PageIndex{53}

If \sin (t)=−\dfrac{1}{4} and t is in Quadrant III, find \cos (t).

Answer

−\dfrac{\sqrt{15}}{4}

54) Find the coordinates of the point on a circle with radius 15 corresponding to an angle of 220°.

55) Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 120°.

Answer

(−10,10\sqrt{3})

56) Find the coordinates of the point on a circle with radius 8 corresponding to an angle of \dfrac{7π}{4}.

57) Find the coordinates of the point on a circle with radius 16 corresponding to an angle of \dfrac{5π}{9}.

Answer

(–2.778,15.757)

58) State the domain of the sine and cosine functions.

59) State the range of the sine and cosine functions.

Answer

[–1,1]

Graphical

For the exercises 60-79, use the given point on the unit circle to find the value of the sine and cosine of t.

60)

CNX_Precalc_Figure_05_02_201.jpg

61)

CNX_Precalc_Figure_05_02_202.jpg

Answer

\sin t=\dfrac{1}{2}, \cos t=−\dfrac{\sqrt{3}}{2}

62)

CNX_Precalc_Figure_05_02_203.jpg

63)

CNX_Precalc_Figure_05_02_204.jpg

Answer

\sin t=− \dfrac{\sqrt{2}}{2}, \cos t=−\dfrac{\sqrt{2}}{2}

64)

CNX_Precalc_Figure_05_02_205.jpg

65)

CNX_Precalc_Figure_05_02_206.jpg

Answer

\sin t=\dfrac{\sqrt{3}}{2},\cos t=−\dfrac{1}{2}

66)

CNX_Precalc_Figure_05_02_207.jpg

67)

CNX_Precalc_Figure_05_02_208.jpg

Answer

\sin t=− \dfrac{\sqrt{2}}{2}, \cos t=\dfrac{\sqrt{2}}{2}

68)

CNX_Precalc_Figure_05_02_209.jpg

69)

CNX_Precalc_Figure_05_02_210.jpg

Answer

\sin t=0, \cos t=−1

70)

CNX_Precalc_Figure_05_02_211.jpg

71)

CNX_Precalc_Figure_05_02_212.jpg

Answer

\sin t=−0.596, \cos t=0.803

72)

CNX_Precalc_Figure_05_02_213.jpg

73)

CNX_Precalc_Figure_05_02_214.jpg

Answer

\sin t=\dfrac{1}{2}, \cos t= \dfrac{\sqrt{3}}{2}

74)

CNX_Precalc_Figure_05_02_215.jpg

75)

CNX_Precalc_Figure_05_02_216.jpg

Answer

\sin t=−\dfrac{1}{2}, \cos t= \dfrac{\sqrt{3}}{2}

76)

CNX_Precalc_Figure_05_02_217.jpg

77)

CNX_Precalc_Figure_05_02_218.jpg

Answer

\sin t=0.761, \cos t=−0.649

78)

CNX_Precalc_Figure_05_02_219.jpg

79)

CNX_Precalc_Figure_05_02_220.jpg

Answer

\sin t=1, \cos t=0

Technology

For the exercises 80-89, use a graphing calculator to evaluate.

80) \sin \dfrac{5π}{9}

81) cos \dfrac{5π}{9}

Answer

−0.1736

82) \sin \dfrac{π}{10}

83) \cos \dfrac{π}{10}

Answer

0.9511

84) \sin \dfrac{3π}{4}

85) \cos \dfrac{3π}{4}

Answer

−0.7071

86) \sin 98°

87) \cos 98°

Answer

−0.1392

88) \cos 310°

89) \sin 310°

Answer

−0.7660

Extensions

For the exercises 90-99, evaluate.

90) \sin \left(\dfrac{11π}{3}\right) \cos \left(\dfrac{−5π}{6}\right)

91) \sin \left(\dfrac{3π}{4}\right) \cos \left(\dfrac{5π}{3}\right)

Answer

\dfrac{\sqrt{2}}{4}

92) \sin \left(− \dfrac{4π}{3}\right) \cos \left(\dfrac{π}{2}\right)

93) \sin \left(\dfrac{−9π}{4}\right) \cos \left(\dfrac{−π}{6}\right)

Answer

−\dfrac{\sqrt{6}}{4}

94) \sin \left(\dfrac{π}{6}\right) \cos \left(\dfrac{−π}{3}\right)

95) \sin \left(\dfrac{7π}{4}\right) \cos \left(\dfrac{−2π}{3}\right)

Answer

\dfrac{\sqrt{2}}{4}

96) \cos \left(\dfrac{5π}{6}\right) \cos \left(\dfrac{2π}{3}\right)

97) \cos \left(\dfrac{−π}{3}\right) \cos \left(\dfrac{π}{4}\right)

Answer

\dfrac{\sqrt{2}}{4}

98) \sin \left(\dfrac{−5π}{4}\right) \sin \left(\dfrac{11π}{6}\right)

99) \sin (π) \sin \left(\dfrac{π}{6}\right)

Answer

0

Real-World Applications

For the exercises 100-104, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0,1), that is, on the due north position. Assume the carousel revolves counter clockwise.

100) What are the coordinates of the child after 45 seconds?

101) What are the coordinates of the child after 90 seconds?

Answer

(0,–1)

102) What is the coordinates of the child after 125 seconds?

103) When will the child have coordinates (0.707,–0.707) if the ride lasts 6 minutes? (There are multiple answers.)

Answer

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

104) When will the child have coordinates (−0.866,−0.5) if the ride last 6 minutes?


5.2E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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