Angles

3. State what a positive or negative angle signifies, and explain how to draw each.

5. Explain the differences between linear speed and angular speed when describing motion along a circular path.

For the following exercises, draw an angle in standard position with the given measure.

7. 300°

9. 135°

11. [latex]\frac{2\pi}{3}[/latex]

13. [latex]\frac{5\pi}{6}[/latex]

15. [latex]-\frac{\pi}{10}[/latex]

17. −120°

19. [latex]\frac{22\pi}{3}[/latex]

21. [latex]-\frac{4\pi}{3}[/latex]

For the following exercises, refer to the figure. Round to two decimal places.

22. Find the arc length.

23. Find the area of the sector.

For the following exercises, refer to the figure. Round to two decimal places.

24. Find the arc length.

25. Find the area of the sector.

For the following exercises, convert angles in radians to degrees.

27. [latex]\frac{\pi}{9}[/latex] radians

29. [latex]\frac{\pi}{3}[/latex] radians

31. [latex]-\frac{5\pi}{12}[/latex] radians

For the following exercises, convert angles in degrees to radians.

33. 90°

35. −540°

39. 150°

For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places.

41. Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of [latex]\frac{\pi}{3}[/latex].

45. Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63°.

For the following exercises, use the given information to find the area of the sector. Round to four decimal places.

47. A sector of a circle has a central angle of 30° and a radius of 20 cm.

49. A sector of a circle with radius of 0.7 inches and an angle of [latex]\pi[/latex] radians.

For the following exercises, find the angle between 0° and 360° that is coterminal to the given angle.

51. −110°

53. 1400°

For the following exercises, find the angle between 0 and [latex]2\pi[/latex] in radians that is coterminal to the given angle.

55. [latex]\frac{10\pi}{3}[/latex]

57. [latex]\frac{44\pi}{9}[/latex]

59. A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

61. A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed $v$, the angular speed in RPM, and the angular speed in deg/s?

65. Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes $(1\text{minute}=\frac{1}{60}\text{degree})$ . The radius of Earth is 3960 miles.

67. Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in $20$ minutes?

73. A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches?

For the following exercises, convert angles in decimal degrees to degrees–minutes–seconds form.

74. 52.375°

75. −73.25°

76. 140.008°

For the following exercises, convert angles in degrees–minutes–seconds form to decimal degrees. Round to four decimal places.

77. 38° 15′ 24″

78. 72° 36″

79. −19° 30′

For the following exercises, find the complement and supplement of the given angle. If a complement or supplement does not exist, state that no complement or supplement exists.

80. 28°

81. 42° 18′ 36″

82. 112°

83. [latex]\frac{\pi}{8}[/latex]

Unit Circle: Sine and Cosine Functions

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.

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

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by [latex]t[/latex] lies.

7. [latex]\sin(t)>0 \text{ and } \cos(t)>0[/latex]

9. [latex]\sin(t)<0 \text{ and } \cos(t)>0[/latex]

For the following exercises, find the exact value of each trigonometric function.

11. [latex]\sin\left(\frac{\pi}{3}\right)[/latex]

13. [latex]\cos\left(\frac{\pi}{3}\right)[/latex]

15. [latex]\cos\left(\frac{\pi}{4}\right)[/latex]

17. [latex]\sin(\pi)[/latex]

21. [latex]\cos\left(\frac{\pi}{6}\right)[/latex]

For the following exercises, state the reference angle for the given angle.

23. [latex]240^\circ[/latex]

27. [latex]135^\circ[/latex]

29. [latex]\frac{2\pi}{3}[/latex]

31. [latex]-\frac{11\pi}{3}[/latex]

For the following exercises, 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.

35. [latex]300^\circ[/latex]

37. [latex]135^\circ[/latex]

43. [latex]\frac{7\pi}{6}[/latex]

45. [latex]\frac{3\pi}{4}[/latex]

For the following exercises, find the requested value.

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

53. If [latex]\sin(t)=-\frac{1}{4}[/latex] and [latex]t[/latex] is in the 3^rd quadrant, find [latex]\cos(t)[/latex].

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of [latex]t[/latex].

61.

63.

65.

67.

69.

73.

For the following exercises, use a graphing calculator to evaluate.

81. [latex]\cos\left(\frac{5\pi}{9}\right)[/latex]

83. [latex]\cos\left(\frac{\pi}{10}\right)[/latex]

87. [latex]\cos\left(98^\circ\right)[/latex]

89. [latex]\sin\left(310^\circ\right)[/latex]

Other Trigonometric Functions

7. [latex]\sec\left(\frac{\pi}{6}\right)[/latex]

13. [latex]\cot\left(\frac{\pi}{4}\right)[/latex]

For the following exercises, use reference angles to evaluate the expression.

19. [latex]\sec\left(\frac{7\pi}{6}\right)[/latex]

21. [latex]\cot\left(\frac{13\pi}{6}\right)[/latex]

33. [latex]\cot\left(240^\circ\right)[/latex]

35. [latex]\sec\left(120^\circ\right)[/latex]

39. If [latex]\cos t=-\frac{1}{3}[/latex], and [latex]t[/latex] is in quadrant III, find [latex]\sin t[/latex], [latex]\sec t[/latex], [latex]\csc t[/latex], [latex]\tan t[/latex], [latex]\cot t[/latex].

41. If [latex]\sin t=\frac{\sqrt{3}}{2}[/latex] and [latex]\cos t=\frac{1}{2}[/latex], find [latex]\sec t[/latex], [latex]\csc t[/latex], [latex]\tan t[/latex], and [latex]\cot t[/latex].

43. If [latex]\sin t=\frac{\sqrt{2}}{2}[/latex], what is [latex]\sin(-t)[/latex]?

45. If [latex]\sec t=3.1[/latex], what is [latex]\sec(-t)[/latex]?

47. If [latex]\tan t=-1.4[/latex], what is [latex]\tan(-t)[/latex]?

For the following exercises, use a graphing calculator to evaluate.

53. [latex]\cot\left(\frac{4\pi}{7}\right)[/latex]

55. [latex]\tan\left(\frac{5\pi}{8}\right)[/latex]

57. [latex]\csc\left(\frac{\pi}{4}\right)[/latex]

61. [latex]\sec\left(310^\circ\right)[/latex]

73. The amount of sunlight in a certain city can be modeled by the function [latex]h=16\cos\left(\frac{1}{500}d\right)[/latex], where [latex]h[/latex] represents the hours of sunlight, and [latex]d[/latex] is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267^th day of the year. State the period of the function.

75. The height of a piston, [latex]h[/latex], in inches, can be modeled by the equation [latex]y=2\cos x+6[/latex], where [latex]x[/latex] represents the crank angle. Find the height of the piston when the crank angle is [latex]55^\circ[/latex].