Math 2312
Test 2 Review
Covering sections 4.5, 4.6, 4.7, 4.8, 5.3, 5.4, 5.5
1.
 x 
Determine the period and amplitude of y  3cos    .
 13 8 
2. Below is the graph of f(x). Find an equation for f(x).
3. Below is the graph of g(x). Find an equation for g(x).
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4. Suppose that yesterday had 12 hours of constant sunlight. The light intensity, I, was at
its highest value of 470 calories/cm2 at midday. If sunrise corresponds to t  0 , find a
formula that models this information.
5. The number of hours of daylight D(t ) at a specific time of year can be modeled by
K
 2

D(t )  sin 
(t  79)   12 , where K is a constant depending upon the locale and t
2
 365

is the number of days, with January 1 corresponding to t  0 . Ignoring the possibility of
a leap year, determine when the longest day of the year will occur if K  8 .
6. Use an inverse function to write  as a function of x.
7.
3

Find the exact value of sin  arctan  .
4

8. An observer is 400 feet from a hot air balloon before it lifts off. Determine the angle of
elevation,  , when the balloon reaches a height of 250 feet. State your answer in
degrees and round to one decimal place.
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9. A granular substance such as sand naturally settles into a cone-shaped pile when poured
from a small aperature. Its height depends on the humidity and adhesion between
granules. The angle of elevation of a pile,  , is called the angle of repose. If the height
of a pile of sand is 15 feet and its diameter is approximately 37 feet, determine the angle
of repose. Round answer to nearest degree.
h

r
10. A communications company erects a 91-foot tall cellular telephone tower on level
ground. Determine the angle of depression,  (in degrees), from the top of the tower to
a point 50 feet from the base of the tower. Round answer to two decimal places.
11. A jet is traveling at 640 miles per hour at a bearing of 48 . After flying for 1.8 hours in
the same direction, how far north will the plane have traveled? Round answer to nearest
mile.
12. While traveling across the flat terrain of Nevada, you notice a mountain directly in front
of you. You calculate that the angle of elevation to the peak is 4.5 , and after you drive
5 miles closer to the mountain it is 7  . Approximate the height of the mountain peak
above your position. Round your answer to the nearest foot.
13. Solve the following equation.
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14. Solve the following equation.
15. Find all solutions of the following equation in the interval  0, 2  .
16. Find all solutions of the following equation in the interval  0, 2  .
17. Solve the multiple-angle equation in the interval  0, 2  .
18. Solve the multiple-angle equation.
19. Use a graphing utility to approximate the solutions (to three decimal places) of the given
  
equation in the interval   ,  .
 2 2
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20. Use inverse functions where needed to find all solutions (if they exist) of the given
equation on the interval  0, 2  .
21. Find the exact value of the given expression using a sum or difference formula.
sin 345
22. Find the exact value of the given expression using a sum or difference formula.
17
cos
12
23. Write the given expression as the cosine of an angle.
cos95 cos 40 + sin 95 sin 40
24. Find the exact value of the given expression.
25.
Find the exact value of cos  u  v  given that sin u  
v are in Quadrant IV.)
26. Simplify the given expression algebraically.
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8
60
and cos v  . (Both u and
17
61
27. Find all solutions of the given equation in the interval  0, 2  .
28. Find the exact solutions of the given equation in the interval  0, 2  .
29. Use a double-angle formula to find the exact value of cos 2u when
7

sin u  , where
u  .
25
2
30. Use the power-reducing formulas to rewrite the given expression in terms of the first
power of the cosine.
31.
Using half-angle formulas, find the exact value of cos
constraints.
4
3
cos x   ,   x 
5
2
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x
given the following
2
32. Find all solutions of the given equation in the interval  0, 2  .
33. Use the sum-to-product formulas to find the exact value of the given expression.
34. Find all solutions of the given equation in the interval  0, 2  .
35. Use the figure below to find the exact value of the given trigonometric expression.
sin

2

10
24
(figure not necessarily to scale)
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Answer Key
1. period: 26 ; amplitude: 3
2.
3.
t 

I  235 1  cos 
6

5. t  170 : June 20
6.
4.
7. 3
5
8.   32.0 
9. 39
10. 61.21
11. 771 miles north
12. 5787 feet
13.
14.
15.
16.
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17.
18.
19.
20.
21. – 3 + 1
2 2
22. – 3 + 1
2 2
23. cos  55 
24.
25.
cos  u  v  
988
1037
26.
27.
28.
29.
cos 2u 
527
625
30.
31.
cos
x
10

2
10
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32.
33.
34.
35.
26
26
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