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Mathematics LibreTexts

6.4E: Exercises

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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 π4 and the terminal side of the angle is in quadrants I and III. Thus, at x=π4,5π4, the sine and cosine values are equal.

2) What would you estimate the cosine of π 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 x2+y2=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) tanπ6

7) secπ6

Answer

233

8) cscπ6

9) cotπ6

Answer

3

10) tanπ4

11) secπ4

Answer

2

12) cscπ4

13) cotπ4

Answer

1

14) tanπ3

15) secπ3

Answer

2

16) cscπ3

17) cotπ3

Answer

33

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

18) tan5π6

19) sec7π6

Answer

233

20) csc11π6

21) cot13π6

Answer

3

22) tan7π4

23) sec3π4

Answer

2

24) csc5π4

25) cot11π4

Answer

1

26) tan8π3

27) sec4π3

Answer

2

28) csc2π3

29) cot5π3

Answer

33

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)

CNX_Precalc_Figure_05_03_201.jpg

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)

CNX_Precalc_Figure_05_03_202.jpg

51)

CNX_Precalc_Figure_05_03_203.jpg

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°.


6.4E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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