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Mathematics LibreTexts

1.E: Trigonometric Functions (Exercises)

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5.1: Angles

In this section, we will examine properties of angles.

Verbal

1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.

Answer

Graph of a circle with an angle inscribed, showing the initial side, terminal side, and vertex.

2) Explain why there are an infinite number of angles that are coterminal to a certain angle.

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

Answer

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

4) How does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph.

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

Answer

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

Graphical

For the exercises 6-21, draw an angle in standard position with the given measure.

6) 30

7) 300

Answer

Graph of a circle with an angle inscribed.

8) 80

9) 135

Answer

Graph of a circle with a 135 degree angle inscribed.

10) 150

11) \dfrac{2π}{3}

Answer

Graph of a circle with a 2pi/3 radians angle inscribed.

12) \dfrac{7π}{4}

13) \dfrac{5π}{6}

Answer

Graph of a circle with 5pi/6 radians angle inscribed.

14) \dfrac{π}{2}

15) −\dfrac{π}{10}

Answer

Graph of a circle with a –pi/10 radians angle inscribed.

16) 415^{\circ}

17) -120^{\circ}

Answer

240^{\circ}

Graph of a circle showing the equivalence of two angles.

18) -315^{\circ}

19)\dfrac{22π}{3}

Answer

\dfrac{4π}{3}

Graph of a circle showing the equivalence of two angles.

20) −\dfrac{π}{6}

21) −\dfrac{4π}{3}

Answer

\dfrac{2π}{3}

Graph of a circle showing the equivalence of two angles.

For the exercises 22-23, refer to Figure below. Round to two decimal places.

Graph of a circle with radius of 3 inches and an angle of 140 degrees.

22) Find the arc length.

23) Find the area of the sector.

Answer

\dfrac{27π}{2}≈11.00 \text{ in}^2

For the exercises 24-25, refer to Figure below. Round to two decimal places.

Graph of a circle with angle of 2pi/5 and a radius of 4.5 cm.

24) Find the arc length.

25) Find the area of the sector.

Answer

\dfrac{81π}{20}≈12.72\text{ cm}^2

Algebraic

For the exercises 26-32, convert angles in radians to degrees.

26) \dfrac{3π}{4} radians

27) \dfrac{π}{9} radians

Answer

20^{\circ}

28) −\dfrac{5π}{4} radians

29) \dfrac{π}{3} radians

Answer

60^{\circ}

30) −\dfrac{7π}{3} radians

31) −\dfrac{5π}{12} radians

Answer

-75^{\circ}

32) \dfrac{11π}{6} radians

For the exercises 33-39, convert angles in degrees to radians.

33) 90^{\circ}

Answer

\dfrac{π}{2} radians

34) 100^{\circ}

35) -540^{\circ}

Answer

−3π radians

36) -120^{\circ}

37) 180^{\circ}

Answer

π radians

38) -315^{\circ}

39) 150^{\circ}

Answer

\dfrac{5π}{6} radians

For the exercises 40-45, use to given information to find the length of a circular arc. Round to two decimal places.

40) Find the length of the arc of a circle of radius 12 inches subtended by a central angle of \dfrac{π}{4} radians.

41) Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of \dfrac{π}{3}.

Answer

\dfrac{5.02π}{3}≈5.26 miles

42) Find the length of the arc of a circle of diameter 14 meters subtended by the central angle of \dfrac{5\pi }{6}.

43) Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of 50^{\circ}.

Answer

\dfrac{25π}{9}≈8.73 centimeters

44) Find the length of the arc of a circle of radius 5 inches subtended by the central angle of 220^{circ}.

45) Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63^{circ}.

Answer

\dfrac{21π}{10}≈6.60 meters

For the exercises 46-49, use the given information to find the area of the sector. Round to four decimal places.

46) A sector of a circle has a central angle of 45^{\circ} and a radius 6 cm.

47) A sector of a circle has a central angle of 30^{\circ} and a radius of 20 cm.

Answer

104.7198\; cm^2

48) A sector of a circle with diameter 10 feet and an angle of \dfrac{π}{2} radians.

49) A sector of a circle with radius of 0.7 inches and an angle of π radians.

Answer

0.7697\; in^2

For the exercises 50-53, find the angle between 0^{\circ} and 360^{\circ} that is coterminal to the given angle.

50) -40^{\circ}

51) -110^{\circ}

Answer

250^{\circ}

52) 700^{\circ}

53) 1400^{\circ}

Answer

320^{\circ}

For the exercises 54-57, find the angle between 0 and 2\pi in radians that is coterminal to the given angle.

54) −\dfrac{π}{9}

55) \dfrac{10π}{3}

Answer

\dfrac{4π}{3}

56) \dfrac{13π}{6}

57) \dfrac{44π}{9}

Answer

\dfrac{8π}{9}

Real-World Applications

58) A truck with 32-inch diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

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?

Answer

1320 rad 210.085 RPM

60) A wheel of radius 8 inches is rotating 15^{\circ}/s. What is the linear speed v, the angular speed in RPM, and the angular speed in rad/s?

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

Answer

7 in./s, 4.77 RPM, 28.65 deg/s

62) A CD has diameter of 120 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed.

63) When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters.

Answer

1,809,557.37 \text{ mm/min}=30.16 \text{ m/s}

64) A person is standing on the equator of Earth (radius 3960 miles). What are his linear and angular speeds?

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

Answer

5.76 miles

66) Find the distance along an arc on the surface of Earth that subtends a central angle of 7 minutes (1 \text{ minute}=\dfrac{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?

Answer

120°

Extensions

68) Two cities have the same longitude. The latitude of city A is 9.00 degrees north and the latitude of city B is 30.00 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities.

69) A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

Answer

794 miles per hour

70) A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

71) Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

Answer

2,234 miles per hour

72) A bicycle has wheels 28 inches in diameter. A tachometer determines that the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.

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

Answer

11.5 inches

74) A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles?

5.2: Unit Circle - Sine and Cosine Functions

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

\textrm{I}

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

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

Answer

\textrm{IV}

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

10) \sin \dfrac{π}{2}

11) \sin \dfrac{π}{3}

Answer

\dfrac{\sqrt{3}}{2}

12) \cos \dfrac{π}{2}

13) \cos \dfrac{π}{3}

Answer

\dfrac{1}{2}

14) \sin \dfrac{π}{4}

15) \cos \dfrac{π}{4}

Answer

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

16) \sin \dfrac{π}{6}

17) \sin π

Answer

0

18) \sin \dfrac{3π}{2}

19) \cos π

Answer

−1

20) \cos 0

21) cos \dfrac{π}{6}

Answer

\dfrac{\sqrt{3}}{2}

22) \sin 0

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 the 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°

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}

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).

53) If \sin (t)=−\dfrac{1}{4} and t is in the 3^{rd} quadrant, 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)

Graph of circle with angle of t inscribed.  Point of (square root of 2 over 2, square root of 2 over 2) is at the intersection of terminal side of angle and edge of circle

61)

Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

Answer

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

62)

Graph of circle with angle of t inscribed. Point of (1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

63)

Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

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

64)

Graph of circle with angle of t inscribed. Point of (1/2, square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

65)

Graph of circle with angle of t inscribed. Point of (-1/2, square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

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

66)

Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

67)

Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

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

68)

Graph of circle with angle of t inscribed. Point of (1,0) is at intersection of terminal side of angle and edge of circle.

69)

Graph of circle with angle of t inscribed. Point of (-1,0) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t=0, \cos t=−1

70)

Graph of circle with angle of t inscribed. Point of (0.111,0.994) is at intersection of terminal side of angle and edge of circle.

71)

Graph of circle with angle of t inscribed. Point of (0.803,-0.596 is at intersection of terminal side of angle and edge of circle.

Answer

\sin t=−0.596, \cos t=0.803

72)

Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

73)

Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

Answer

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

74)

Graph of circle with angle of t inscribed. Point of (negative square root of 3 over 2, -1/2) is at intersection of terminal side of angle and edge of circle.

75)

Graph of circle with angle of t inscribed. Point of (square root of 3 over 2, -1/2) is at intersection of terminal side of angle and edge of circle.

Answer

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

76)

Graph of circle with angle of t inscribed. Point of (0, -1) is at intersection of terminal side of angle and edge of circle.

77)

Graph of circle with angle of t inscribed. Point of (-0.649, 0.761) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t=0.761, \cos t=−0.649

78)

Graph of circle with angle of t inscribed. Point of (-0.948, -0.317) is at intersection of terminal side of angle and edge of circle.

79)

Graph of circle with angle of t inscribed. Point of (0, 1) is at intersection of terminal side of angle and edge of circle.

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?

5.3: The Other Trigonometric Functions

Verbal

1) On an interval of [ 0,2π ), can the sine and cosine values of a radian measure ever be equal? If so, where?

Answer

Yes, when the reference angle is \dfrac{π}{4} and the terminal side of the angle is in quadrants I and III. Thus, at x=\dfrac{π}{4},\dfrac{5π}{4}, the sine and cosine values are equal.

2) What would you estimate the cosine of \pi degrees to be? Explain your reasoning.

3) For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Answer

Substitute the sine of the angle in for y in the Pythagorean Theorem x^2+y^2=1. Solve for x and take the negative solution.

4) Describe the secant function.

5) Tangent and cotangent have a period of π. What does this tell us about the output of these functions?

Answer

The outputs of tangent and cotangent will repeat every π units.

Algebraic

For the exercises 6-17, find the exact value of each expression.

6) \tan \dfrac{π}{6}

7) \sec \dfrac{π}{6}

Answer

\dfrac{2\sqrt{3}}{3}

8) \csc \dfrac{π}{6}

9) \cot \dfrac{π}{6}

Answer

\sqrt{3}

10) \tan \dfrac{π}{4}

11) \sec \dfrac{π}{4}

Answer

\sqrt{2}

12) \csc \dfrac{π}{4}

13) \cot \dfrac{π}{4}

Answer

1

14) \tan \dfrac{π}{3}

15) \sec \dfrac{π}{3}

Answer

2

16) \csc \dfrac{π}{3}

17) \cot \dfrac{π}{3}

Answer

\dfrac{\sqrt{3}}{3}

For the exercises 18-48, use reference angles to evaluate the expression.

18) \tan \dfrac{5π}{6}

19) \sec \dfrac{7π}{6}

Answer

−\dfrac{2\sqrt{3}}{3}

20) \csc \dfrac{11π}{6}

21) \cot \dfrac{13π}{6}

Answer

\sqrt{3}

22) \tan \dfrac{7π}{4}

23) \sec \dfrac{3π}{4}

Answer

−\sqrt{2}

24) \csc \dfrac{5π}{4}

25) \cot \dfrac{11π}{4}

Answer

−1

26) \tan \dfrac{8π}{3}

27) \sec \dfrac{4π}{3}

Answer

−2

28) \csc \dfrac{2π}{3}

29) \cot \dfrac{5π}{3}

Answer

−\dfrac{\sqrt{3}}{3}

30) \tan 225°

31) \sec 300°

Answer

2

32) \csc 150°

33) \cot 240°

Answer

\dfrac{\sqrt{3}}{3}

34) \tan 330°

35) \sec 120°

Answer

−2

36) \csc 210°

37) \cot 315°

Answer

−1

38) If \sin t= \dfrac{3}{4}, and t is in quadrant II, find \cos t, \sec t, \csc t, \tan t, \cot t .

39) If \cos t=−\dfrac{1}{3}, and t is in quadrant III, find \sin t, \sec t, \csc t, \tan t, \cot t.

Answer

If \sin t=−\dfrac{2\sqrt{2}}{3}, \sec t=−3, \csc t=−\csc t=−\dfrac{3\sqrt{2}}{4},\tan t=2\sqrt{2}, \cot t= \dfrac{\sqrt{2}}{4}

40) If \tan t=\dfrac{12}{5}, and 0≤t< \dfrac{π}{2}, find \sin t, \cos t, \sec t, \csc t, and \cot t.

41) If \sin t= \dfrac{\sqrt{3}}{2} and \cos t=\dfrac{1}{2}, find \sec t, \csc t, \tan t, and \cot t.

Answer

\sec t=2, \csc t=\csc t=\dfrac{2\sqrt{3}}{3}, \tan t= \sqrt{3}, \cot t= \dfrac{\sqrt{3}}{3}

42) If \sin 40°≈0.643 \; \cos 40°≈0.766 \; \sec 40°,\csc 40°,\tan 40°, \text{ and } \cot 40°.

43) If \sin t= \dfrac{\sqrt{2}}{2}, what is the \sin (−t)?

Answer

−\dfrac{\sqrt{2}}{2}

44) If \cos t= \dfrac{1}{2}, what is the \cos (−t)?

45) If \sec t=3.1, what is the \sec (−t)?

Answer

3.1

46) If \csc t=0.34, what is the \csc (−t)?

47) If \tan t=−1.4, what is the \tan (−t)?

Answer

1.4

48) If \cot t=9.23, what is the \cot (−t)?

Graphical

For the exercises 49-51, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

49)

This is an image of a graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t= \dfrac{\sqrt{2}}{2}, \cos t= \dfrac{\sqrt{2}}{2}, \tan t=1,\cot t=1,\sec t= \sqrt{2}, \csc t= \csc t= \sqrt{2}

50)

This is an image of a graph of circle with angle of t inscribed. Point of (square root of 3 over 2, 1/2) is at intersection of terminal side of angle and edge of circle.

51)

This is an image of a graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

Answer

\sin t=−\dfrac{\sqrt{3}}{2}, \cos t=−\dfrac{1}{2}, \tan t=\sqrt{3}, \cot t= \dfrac{\sqrt{3}}{3}, \sec t=−2, \csc t=−\csc t=−\dfrac{2\sqrt{3}}{3}

Technology

For the exercises 52-61, use a graphing calculator to evaluate.

52) \csc \dfrac{5π}{9}

53) \cot \dfrac{4π}{7}

Answer

–0.228

54) \sec \dfrac{π}{10}

55) \tan \dfrac{5π}{8}

Answer

–2.414

56) \sec \dfrac{3π}{4}

57) \csc \dfrac{π}{4}

Answer

1.414

58) \tan 98°

59) \cot 33°

Answer

1.540

60) \cot 140°

61) \sec 310°

Answer

1.556

Extensions

For the exercises 62-69, use identities to evaluate the expression.

62) If \tan (t)≈2.7, and \sin (t)≈0.94, find \cos (t).

63) If \tan (t)≈1.3, and \cos (t)≈0.61, find \sin (t).

Answer

\sin (t)≈0.79

64) If \csc (t)≈3.2, and \csc (t)≈3.2, and \cos (t)≈0.95, find \tan (t).

65) If \cot (t)≈0.58, and \cos (t)≈0.5, find \csc (t).

Answer

\csc (t)≈1.16

66) Determine whether the function f(x)=2 \sin x \cos x is even, odd, or neither.

67) Determine whether the function f(x)=3 \sin ^2 x \cos x + \sec x is even, odd, or neither.

Answer

even

68) Determine whether the function f(x)= \sin x −2 \cos ^2 x is even, odd, or neither.

69) Determine whether the function f(x)= \csc ^2 x+ \sec x is even, odd, or neither.

Answer

even

For the exercises 70-71, use identities to simplify the expression.

70) \csc t \tan t

71) \dfrac{\sec t}{ \csc t}

Answer

\dfrac{ \sin t}{ \cos t}= \tan t

Real-World Applications

72) The amount of sunlight in a certain city can be modeled by the function h=15 \cos \left(\dfrac{1}{600}d\right), where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42^{nd} day of the year. State the period of the function.

73) The amount of sunlight in a certain city can be modeled by the function h=16 \cos \left(\dfrac{1}{500}d\right), where h represents the hours of sunlight, and d 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.

Answer

13.77 hours, period: 1000π

74) The equation P=20 \sin (2πt)+100 models the blood pressure, P, where t represents time in seconds.

  1. Find the blood pressure after 15 seconds.
  2. What are the maximum and minimum blood pressures?

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

Answer

7.73 inches

76) The height of a piston, h,in inches, can be modeled by the equation y=2 \cos x+5, where x represents the crank angle. Find the height of the piston when the crank angle is 55°.

5.4: Right Triangle Trigonometry

Verbal

1) For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

A right triangle.

Answer

A right triangle with side opposite, adjacent, and hypotenuse labeled.

2) When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the x- and y-coordinates?

3) The tangent of an angle compares which sides of the right triangle?

Answer

The tangent of an angle is the ratio of the opposite side to the adjacent side.

4) What is the relationship between the two acute angles in a right triangle?

5) Explain the cofunction identity.

Answer

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

Algebraic

For the exercises 6-9, use cofunctions of complementary angles.

6) \cos (34°)= \sin (\_\_°)

7) \cos (\dfrac{π}{3})= \sin (\_\_\_)

Answer

\dfrac{π}{6}

8) \csc (21°) = \sec (\_\_\_°)

9) \tan (\dfrac{π}{4})= \cot (\_\_)

Answer

\dfrac{π}{4}

For the exercises 10-16, find the lengths of the missing sides if side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse.

10) \cos B= \dfrac{4}{5},a=10

11) \sin B= \dfrac{1}{2}, a=20

Answer

b= \dfrac{20\sqrt{3}}{3},c= \dfrac{40\sqrt{3}}{3}

12) \tan A= \dfrac{5}{12},b=6

13) \tan A=100,b=100

Answer

a=10,000,c=10,000.5

14) \sin B=\dfrac{1}{\sqrt{3}}, a=2

15) a=5, ∡ A=60^∘

Answer

b=\dfrac{5\sqrt{3}}{3},c=\dfrac{10\sqrt{3}}{3}

16) c=12, ∡ A=45^∘

Graphical

For the exercises 17-22, use Figure below to evaluate each trigonometric function of angle A.

A right triangle with sides 4 and 10 and angle of A labeled which is opposite the side labeled 10.

17) \sin A

Answer

\dfrac{5\sqrt{29}}{29}

18) \cos A

19) \tan A

Answer

\dfrac{5}{2}

20) \csc A

21) \sec A

Answer

\dfrac{\sqrt{29}}{2}

22) \cot A

For the exercises 23-,28 use Figure below to evaluate each trigonometric function of angle A.

A right triangle with sides of 10 and 8 and angle of A labeled which is opposite the side labeled 10.

23) \sin A

Answer

\dfrac{5\sqrt{41}}{41}

24) \cos A

25) \tan A

Answer

\dfrac{5}{4}

26) \csc A

27) \sec A

Answer

\dfrac{\sqrt{41}}{4}

28) \cot A

For the exercises 29-31, solve for the unknown sides of the given triangle.

29)

A right triangle with sides of 7, b, and c labeled. Angles of B and 30 degrees also labeled.  The 30 degree angle is opposite the side labeled 7.

Answer

c=14, b=7\sqrt{3}

30)

A right triangle with sides of 10, a, and c. Angles of 60 degrees and A also labeled.  The 60 degree angle is opposite the side labeled 10.

31)

A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.

Answer

a=15, b=15

Technology

For the exercises 32-41, use a calculator to find the length of each side to four decimal places.

32)

A right triangle with sides of 10, a, and c. Angles of A and 62 degrees are also labeled.  The 62 degree angle is opposite the side labeled 10.

33)

A right triangle with sides of 7, b, and c. Angles of 35 degrees and B are also labeled.

Answer

b=9.9970, c=12.2041

34)

A right triangle with sides of a, b, and 10 labeled. Angles of 65 degrees and B are also labeled.

35)

A right triangle with sides a, b, and 12. Angles of 10 degrees and B are also labeled.

Answer

a=2.0838, b=11.8177

36)

A right triangle with corners labeled A, B, and C. Sides labeled b, c, and 16.5. Angle of 81 degrees also labeled.

37) b=15, ∡B=15^∘

Answer

a=55.9808,c=57.9555

38) c=200, ∡B=5^∘

39) c=50, ∡B=21^∘

Answer

a=46.6790,b=17.9184

40) a=30, ∡A=27^∘

41) b=3.5, ∡A=78^∘

Answer

a=16.4662,c=16.8341

Extensions

42) Find x.

A triangle with angles of 63 degrees and 39 degrees and side x. Bisector in triangle with length of 82.

43) Find x.

A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.

Answer

188.3159

44) Find x.

A right triangle with side of 115 and angle of 35 degrees. Within right triangle there is another right triangle with angle of 56 degrees. Side length difference between two triangles is x.

45) Find x.

A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is x.

Answer

200.6737

46) A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36°, and that the angle of depression to the bottom of the tower is 23°. How tall is the tower?

47) A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 43°, and that the angle of depression to the bottom of the tower is 31°. How tall is the tower?

Answer

498.3471 ft

48) A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 15°, and that the angle of depression to the bottom of the tower is . How far is the person from the monument?

49) A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18°, and that the angle of depression to the bottom of the monument is . How far is the person from the monument?

Answer

1060.09 ft

50) There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be 40°. From the same location, the angle of elevation to the top of the antenna is measured to be 43°. Find the height of the antenna.

51) There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be 36°. From the same location, the angle of elevation to the top of the lightning rod is measured to be 38°. Find the height of the lightning rod.

Answer

27.372 ft

Real-World Applications

52) A 33-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?

53) A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?

Answer

22.6506 ft

54) The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

55) The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

Answer

368.7633 ft

56) Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60°, how far from the base of the tree am I?


This page titled 1.E: Trigonometric Functions (Exercises) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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