5.3E: 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
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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
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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
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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
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2√33
8) cscπ6
9) cotπ6
- Answer
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√3
10) tanπ4
11) secπ4
- Answer
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√2
12) cscπ4
13) cotπ4
- Answer
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1
14) tanπ3
15) secπ3
- Answer
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2
16) cscπ3
17) cotπ3
- Answer
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√33
For the exercises 18-48, use reference angles to evaluate the expression.
18) tan5π6
19) sec7π6
- Answer
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−2√33
20) csc11π6
21) cot13π6
- Answer
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√3
22) tan7π4
23) sec3π4
- Answer
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−√2
24) csc5π4
25) cot11π4
- Answer
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−1
26) tan8π3
27) sec4π3
- Answer
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−2
28) csc2π3
29) cot5π3
- Answer
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−√33
30) tan225°
31) sec300°
- Answer
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2
32) csc150°
33) cot240°
- Answer
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√33
34) tan330°
35) sec120°
- Answer
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−2
36) csc210°
37) cot315°
- Answer
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−1
38) If sint=34, and t is in quadrant II, find cost,sect,csct,tant,cott.
39) If cost=−13, and t is in quadrant III, find sint,sect,csct,tant,cott.
- Answer
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If sint=−2√23,sect=−3,csct=−csct=−3√24,tant=2√2,cott=√24
40) If tant=125, and 0≤t<π2, find sint,cost,sect,csct, and cott.
41) If sint=√32 and cost=12, find sect,csct,tant, and cott.
- Answer
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sect=2,csct=csct=2√33,tant=√3,cott=√33
42) If sin40°≈0.643cos40°≈0.766sec40°,csc40°,tan40°, and cot40°.
43) If sint=√22, what is the sin(−t)?
- Answer
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−√22
44) If cost=12, what is the cos(−t)?
45) If sect=3.1, what is the sec(−t)?
- Answer
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3.1
46) If csct=0.34, what is the csc(−t)?
47) If tant=−1.4, what is the tan(−t)?
- Answer
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1.4
48) If cott=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)
- Answer
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sint=√22,cost=√22,tant=1,cott=1,sect=√2,csct=csct=√2
50)
51)
- Answer
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sint=−√32,cost=−12,tant=√3,cott=√33,sect=−2,csct=−csct=−2√33
Technology
For the exercises 52-61, use a graphing calculator to evaluate.
52) csc5π9
53) cot4π7
- Answer
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–0.228
54) \sec \dfrac{π}{10}
55) \tan \dfrac{5π}{8}
- Answer
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–2.414
56) \sec \dfrac{3π}{4}
57) \csc \dfrac{π}{4}
- Answer
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1.414
58) \tan 98°
59) \cot 33°
- Answer
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1.540
60) \cot 140°
61) \sec 310°
- Answer
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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
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\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
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\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
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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
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even
For the exercises 70-71, use identities to simplify the expression.
70) \csc t \tan t
71) \dfrac{\sec t}{ \csc t}
- Answer
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\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
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13.77 hours, period: 1000π
74) The equation P=20 \sin (2πt)+100 models the blood pressure, P, where t represents time in seconds.
- Find the blood pressure after 15 seconds.
- 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
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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°.