PRECALCULUS A Semester Exam Review
The semester A examination for Precalculus consists of two parts. Part 1 is selected response on which a calculator will not be allowed. Part 2 is short answer on which a calculator will be allowed.
Pages with the symbol indicate that a student should be prepared to complete questions like these with or without a calculator.
The formulas below are provided in the examination booklet.
Trigonometric Identities: sin
2   cos
2  
1 sec 2

2
 csc 2

2
 sin
          sin cos

2

2

2 cos
         tan sin 2
  
 
 
 tan

1 tan


 tan tan

 cos 2
  cos 2 tan 2
 
  sin 2
2 tan

2 
 
2 cos 2

2
 tan
2 sin

2




Law of Cosines and Sines:
Law of Cosines: c
2  a
2  b
2 
2 ab cos C
Law of Sines: sin A a
Area of a Triangle:
 sin B b
 sin C c
Area =
1
2 ab sin C
Area
   a
   s
 c

, where s

1 
2
Length of Arc: s
 r

,

in radians s


360

2
 r

,

in degrees
   sin


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2.
PRECALCULUS A
PART 1 NO CALCULATOR SECTION
1. Sketch the graph of the piece-wise function
Look at the graph of the piecewise function below. y
 
 x
2 x
, if
 x
1, if
 x
0

0
Semester Exam Review x
Which of the following functions is represented by the graph?
A
 
 x
2
, if x , if x x


0
0
B
C
D
 
 x
2
, if x , if x x


0
0
 
 x

, if x 2 , if x x


0
0


 x
2 x , if
, if x x


0
0
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6.
3.
PRECALCULUS A
Look at the graph of the piecewise function below.
y
-2 2 4 x
What type of discontinuity does the graph have at the following x values? a. x
 
2 b. x

2 c. x

4
Semester Exam Review
4. Let

 cx x
2

7, if x

5
, if x

5
.
What is the value of c that will make
 
continuous at x

5 ?
5. Let

4

, if x

11
5 x

2, if x

11
.
What is the value of c that will make
  continuous at x

11 ?
Which of the following is true about the function
 x x

4

3
?
A
B
C
D
The function is continuous for all real numbers.
The function is discontinuous at x

3 only.
The function is discontinuous at x
 
4 only.
The function is discontinuous at x

3 and x
 
4 .
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PRECALCULUS A
7. Look at the graph of a function below. y
Semester Exam Review x
8. Look at the graph of a function below.
Does the graph above represent an odd function, an even function, or a function that is neither odd nor even? y x
9.
Does the graph above represent an odd function, an even function, or a function that is neither odd nor even?
Determine whether each function below is even, odd, or neither even nor odd. a.
 sin x
 x
3 b. c.
 
 

 x
2 cos
 x
4

4 x
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PRECALCULUS A Semester Exam Review
10. If
2
   x 3 , which of the following statements is NOT true?
A
B
The graph of is symmetric with respect to the is an even function.
C The range of is all real numbers.
D As x
 
,
 
11. Look at the graph of the function below. y y -axis. x a.
What is the domain of this function? _______________________ b.
What is the range of this function? _________________________
12. For each function below, find a formula for f

1
 
and state any restrictions on the domain. a.
 x

2 b.
   x
3 
4
13. True or false. a.
The function
  
5
  
2 represents a vertical stretch of the graph of by a factor of 5, followed by a vertical translation down 2 units. b.
The function
  
7 f represents a vertical and horizontal shrinking of the graph of ( ) .
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PRECALCULUS A Semester Exam Review
14. Match the transformations that would create the graph of
 
.
______
  
3
  from the graph of
A Stretch the graph of ( ) horizontally.
______
   f
 
______
 

1 g x f x

3


______
  
1
3
 
B Stretch the graph of ( ) vertically.
C Shrink the graph of ( ) horizontally.
D Shrink the graph of ( ) vertically.
For items 15 and 16, use the graphs of y
  
and below. y
   x x
15. Which of the following represents the relationship between
A
  
2
 
1

B

1
2
 
1

C
   f
  
1
D
   f


1
2 x



1
16. Sketch the graph of y
  
.
and ?
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PRECALCULUS A Semester Exam Review
17. Write the definitions of the six circular functions for an angle

in standard position in terms of r
18. If sin
  
4
with
5 cos
 
0 , what are the values of the other five trigonometric functions?
19. For each of the following, write the quadrant in which the terminal side of

lies. a. sin
 
0, tan
 
0 b. cos
 
0, tan
 
0 c. sec
 
0, csc
 
0
20. Convert to radian measure. Leave your answer in terms of

. a. 40 o b. 165 o
21. The coordinates of point A are
 
B are

0.8, 0.6
 as shown below. Find the value of the following.
B a. sin
 b. cos
 c. tan

0

A
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PRECALCULUS A Semester Exam Review
22. Determine the exact value of the following. a. sin b.

5
  cos

4
 c. d.

3
  sin

2
 e. cos
  f. g. tan

7

4 h. cos

4

3
i.

5
  tan

3
 tan sin j sin k.

5
  cos

6
 l.

7
  tan

6
 m. sec

5

4 n.

5
  cot
 6  o.

4
  csc
 3 
23. Sketch the graphs of the six circular functions on the interval

2
 x 2

24. Which of the following functions does NOT have a period of 2

?
A
   sin x

11

6
B
   tan x
C
   sec x
D
   csc x
25. What is the period of y

?
26. What is the value of b such that y
 cos has a period of

3
?
27. What is the value of c such that y
 sec has a period of 10?
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PRECALCULUS A Semester Exam Review
28. a. Write an equation for each inverse function graph (i-iii). i. ii. iii. y y y

2 

2 
2 x x
 
2
 
2 x b. Use the following intervals to complete the table below.
[–1,1] (
 
  

 
,
2 2



1
Sin x

1
Cos x

1
Tan x
Domain
Range
29. Determine the exact value of the following. a. Sin

1
1
  b. Cos

1



1
2

 c. Tan

1
  d. Sin

1
  e. Cos

1
  f. Tan

1
 

 
,
2 2
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PRECALCULUS A
30. Determine the exact value of the following. a.
 cos Sin


1


2
3



 b.


1
   c.
 tan Cos

1



1
2




31. Determine the exact value of the following. a.
 sin Csc

1
8
 

 b.
 tan Sin

1

12


13


 c.

1

Cos cos

11

6


 d.
 sin Tan

1



4
3




32. Write a sinusoidal equation for the following graphs
32a. y
32b. y
Semester Exam Review x

2

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2
 x
10

PRECALCULUS A y
32c.

6
7

6
13

6 x
Semester Exam Review
33. Determine the equation that best describes a sine curve with amplitude 3, period
 of 6, and a phase shift of to the right.
2
34. State the amplitude, period, the phase shift and vertical translation of the sinusoid relative to the basic function
   sin x or
   cos x . Sketch the graph, marking the x - and y -axes appropriately. a.
  

2sin 3
  x


6




5 b.
 
5cos

  x

1
  c.
  x
   
2 .
35. Simplify the following expressions and evaluate. a.
5
 sin cos
8
3

8

5
 cos sin
8
3

8 b.
5
 cos cos
6

6

5
 sin sin
6

6 c. tan 32 o  tan 28 o
 o
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PRECALCULUS A
36. If cot A

12
5
and 0 a. sin 2 A b. cos 2 A c. sin
Semester Exam Review
A

2
, determine the exact value of the following. d. cos
37. Prove the following identities. a.
   cos
 b.
 sin x
 cos x

2 x c. csc x
2 x
 sin x d. sin cos


 cos sin


   e. sin
 x
 y
  sin
 x
 y
 
2sin cos y f. sin
2
  sin
2
 tan
2
  tan
2

38. Solve the following equations on the interval 0 o a. 2 sin
  
2 b. 3 cos

4 5 cos
 
5
39. Solve the following equations on the interval 0 a. tan x 1 0 b. 2 sin
2 x

3sin x
 
0
360 o
. x 2

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PRECALCULUS A Semester Exam Review
PART 2 CALCULATOR SECTION
A calculator may be used on items 40 through 56. Make sure that your calculator is in the appropriate mode (radian or degree) for each item. Round to the number of decimal places specified in each item.
40. Complete the following chart using s
 r

.
Radius
6 inches
Angle(radians) Arc Length

5
4

6
15
 feet
10 meters 30 meters
41. A ball on a string is swinging from the ceiling, as shown in the figure below.
Let d d
represent the distance that the ball is from the wall at time
Assume that d varies sinusoidally with time. t .
When
When 3
When t t t



0
6 seconds, the ball is farthest from the wall, seconds, the ball is closest to the wall, d
 d

20
160 cm. seconds, the ball is again farthest from the wall, d cm.

160 cm. a. Sketch a graph of d as a function of time. b. Write a trigonometric function for d as a function of time. c. What is the distance of the ball from the wall at t

5 seconds? d. What is the value of t the first time the ball is 40 cm from the wall? Your answer should be correct to three places after the decimal point.
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PRECALCULUS A
MCPS © 2012–2013
Semester Exam Review
42. Sara is riding a Ferris wheel. Her sister, Kari, starts a stopwatch and records some data. Let h represent Sara’s height above the ground at time t. Kari notices that
Sara is at the highest point, 80 feet above the ground, when t

3 seconds. When t

7 seconds Sara is at the lowest point, 20 feet above the ground. Assume that the height h varies sinusoidally with time t . a.
Write a trigonometric equation for the height h of Sara above the ground as a function of time t . b.
What will the height of Sara be above the ground at t

11.5
seconds?
Your answer should be correct to three places after the decimal point. c.
Determine the first two times, t

0 , when the height of Sara above the ground is 70 feet. Your answer should be correct to three places after the decimal point.
43. At Ocean Tide Dock the first low tide of the day occurs at midnight, when the depth of the water is 2 meters, and the first high tide occurs at 6:30 A.M. with a depth of 8 meters. Assume that the depth of the water varies sinusoidally with time. a.
Sketch and label a graph showing the depth ( d ) of the water at the dock as a function of the number of hours after midnight ( t ). b.
Determine a trigonometric model that represents the graph. c.
Suppose a tanker requiring at least 3 meters of water depth is planning to dock after midnight. Determine the earliest possible time that the tanker can dock.
44. Solve for

, where 0 o tenth of a degree. a. 3cos
  
7 b.
360 o
. Your answer should be correct to the nearest
3sin
2
 
7sin
  
0
45. How many triangles ABC are possible if

A = 20 o
, b

40 , and a

10 ?
46. Given ABC , where measure of side b
A 41 o
, 58 o
, and c

19.7
cm, determine the
. Your answer should be correct to three places after the decimal point.
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PRECALCULUS A Semester Exam Review
47. In

ABC , a

9, b

12, c

16 . What is the measure of

B ? Your answer should be correct to the nearest tenth of a degree.
48. Determine the remaining sides and angles of a triangle with

A side a

11.4
and side 12.8
= 58 o
, b

. Your answers (sides and angles) should correct to the nearest tenth.
49. From a point 200 feet from its base, the angle of elevation from the ground to the top of a lighthouse is 55 degrees. How tall is the lighthouse? Your answer should be correct to three places after the decimal point.
50. A truck is traveling down a mountain. A sign says that the degree of incline is 7 degrees. After the truck has traveled one mile (5280 feet), how many feet in elevation has the truck fallen? Your answer should be correct to three places after the decimal point.
51. A person is walking towards a mountain. At one point, the angle of elevation from the ground to the top of the mountain is 37 degrees. After walking another 1000 feet, the angle of elevation from the ground to the top of the mountain is 40 degrees. How high is the mountain? Your answer should be correct to three places after the decimal point.
52. The owner of the garage shown below plans to install a trim along the roof. The lengths required are in bold. How many feet of trim should be purchased? Your answer should be correct to three places after the decimal point.
50 o 50 o
20 feet
53. An airplane needs to take a detour around a group of thunderstorms, as shown in the figure below. How much farther does the plane have to travel due to the detour? Your answer should be correct to three places after the decimal point.
34 o 50 miles
20 o
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PRECALCULUS A Semester Exam Review
54. Determine the area of

ABC if a

4, b

10,
 
30 o
55 . A real estate appraiser wishes to find the value of the lot below.
160 ft
62 o
250 ft a.
Find the area of the lot. Your answer should be correct to three places after the decimal point. b.
An acre is 43560 square feet. If land is valued at $56,000 per acre, how much is the land worth? Your answer should be correct to the nearest dollar.
56. Find the area of the quadrilateral below. Your answer should be correct to three places after the decimal point.
35
126 o
38
50
72 o
58
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