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Name: ________________________ Class: ___________________ Date: __________
Ma1316x1R
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. Using trigonometric identities, determine
a.
which of the following is equivalent to the
following expression.
tan θ + cot θ
a. cos θ + sec θ b. 1 c. sin θ
d. sec θ + csc θ e. csc θ sec θ
2
2. Determine which of the graphs below
represents
b.
|x sinx |
c.
1
ID: A
Name: ________________________
ID: A
d.
e.
3. Which of the following can be inserted to make the statement true?
arccos
a. x b.
4−x
2
2
= arcsinÊÁË
8 − x2
x
c.
2
x
x
d.
2
2
ˆ˜ , 0 ≤ x ≤ 2
¯
e. 4 − x
2
2
Name: ________________________
ID: A
4. Which of the following functions is represented
5. Which of the following functions is represented
by the graph below?
a. 2 arcsin
x
2
d. arcsin2x
by the graph below?
b. arccos(x + 1)
e. arccos
ÊÁ x + 1 ˆ˜
˜˜˜ b. 1 arcsin x
˜
2
4
Ë 2 ¯
ÊÁ x ˆ˜ π
c. arccos ÁÁÁÁ ˜˜˜˜ −
d. 2 arccos(x + 1)
Ë4 ¯ 2
1
arcsin2x
e.
2
a. arccos ÁÁÁ
Á
c. 2 arccos x
x
2
Short Answer
6. Determine the quadrant in which a −128° 41'
13. Rewrite −
angle lies.
14. Rewrite −219.80° in radian measure. Round to
7. Determine two coterminal angles (one positive
and one negative) for θ = −489° .
three decimal places.
15. Find the angle, in radians, in the figure below if
8. Convert 329.474° to D° M' S" form.
S = 10 and r = 7.
9. Find (if possible) the supplement of 118°.
10. Determine the quadrant in which an angle, θ,
lies if θ = 4.90 radians.
11. Determine two coterminal angles (one positive
and one negative) for θ =
3π
.
4
12. Find (if possible) the complement of
7π
in degree measure.
18
π
7
.
3
Name: ________________________
ID: A
16. Find the length of the arc, S, on a circle of
3
1
and cos 30° =
,
2
2
determine the following:
21. Given sin30° =
radius 3 meters intercepted by a central angle
of 210° . Round to two decimal places.
tan30°
17. Find the area of the sector of the circle with
radius 2 meters and central angle
11π
.
6
ÁÊ π
ÁË 8
22. Use a calculator to evaluate csc ÁÁÁÁ
18. A car is traveling along Route 66 at a rate of 75
your answer to four decimal places.
miles per hour, and the diameter of its wheels
are 2.4 feet. Find the number of revolutions per
minute the wheels are turning. Round answer
to one decimal place.
3
, find the value of θ in degrees
2
(0 < θ < 90° ) without the aid of a calculator.
23. If sinθ =
19. Find the exact value of csc θ , using the triangle
24. Using the figure below, if θ = 34° and y = 7,
shown in the figure below, if a = 7 and b = 24.
determine the exact value of x.
20. If θ is an acute angle and cot θ =
sin θ .
˜ˆ˜
˜˜˜ . Round
¯
1
, determine
3
4
Name: ________________________
ID: A
25. Will Barrow wanted to know how tall the flagpole was in front of his school. To find its height, he drove
a stake into the ground at the tip of the flagpole's shadow and recorded the angle of elevation at two
different times during the day. He then measured the distance between the stakes. Will's data is below:
Stake
Time
A
2:00 PM
B
3:00 PM
Distance between stakes A & B
Angle of
Elevation
79°
58°
10 feet
Determine the height of the flagpole. Round your answer to nearest foot.
27. The point (−5, − 12) is on the terminal side of
26. Given the figure below, determine the value of
sin θ .
an angle in standard position. Determine the
exact value of secθ .
28. State the quadrant in which θ lies if tanθ > 0
and cosθ > 0.
29. Determine the exact value of cscθ when
cos θ =
5
and cscθ > 0.
13
30. The terminal side of θ lies on the line
7x + 24y = 0 in the second quadrant. Find the
exact value of tanθ .
31. Determine the exact value of the cosecant of
the quadrant angle
5
π
2
.
Name: ________________________
ID: A
32. Determine the exact value of sin (−315° ) .
37. Determine the period and amplitude of the
following function.
ÊÁ 3x π ˆ˜
y = 4 cos ÁÁÁÁ
+ ˜˜˜˜
4¯
Ë 4
33. Use a calculator to evaluate tan335° . Round
your answer to four decimal places.
34. Given the equation below, determine two
solutions such that 0 ≤ θ < 2π .
sec θ = −2
35. Find the point ÊÁË x, y ˆ˜¯ on the unit circle that
corresponds to the real number
5π
. Use your
6
results to evaluate cos t .
36. The displacement from equilibrium of an
oscillating weight suspended by a spring is
given by y(t) = 2 cos 8t , where y is the
displacement in centimeters and t is the time in
seconds. Find the displacement when t = 0.45,
rounding answer to four decimal places.
38. Describe the relationship between f(x) = cos(x)
and g(x) = cos 7x − 11 . Consider amplitude,
period, and shifts.
39. Sketch the graph of the function below, being
sure to include at least two full periods.
ÁÊ
π ˜ˆ
y = 2 cos ÁÁÁÁ x − ˜˜˜˜
2 ¯
Ë
40. Use a graphing utility to graph the function below. Be sure to include at least two full periods.
y = 3 cos (x + 3π ) + 2
6
Name: ________________________
ID: A
41. Find a, b, and c for the function
42. Which of the following functions is represented
f(x) = a cos (bx − c) such that the graph of f(x)
matches the graph below.
by the graph below?
43. Sketch the graph of the given function. Make
sure to include at least two periods.
y = −3 sec (x + π )
44. Use a graphing utility to graph the function below, making sure to show at least two periods.
tan
x
4
45. Use a graphing utility to graph the expression below, making sure to show at least two periods.
1
csc (x − π )
2
7
Name: ________________________
ID: A
46. Use a graphing utility to graph the function. Describe the behavior of the function as x approaches 0.
f(x) =
3
+ cos 2x, x > 0
x
47. Evaluate arccos
50. Use an inverse function to write θ as a function
3
without using a calculator.
2
of x.
ˆ
ÊÁ
3 ˜˜˜˜
˜˜ without using a
48. Evaluate tan ÁÁÁ −
ÁÁ 3 ˜˜˜
¯
Ë
calculator.
ÁÁ
−1 Á
−1
(−0.87) .
Round your answer to two decimal places.
49. Use a calculator to evaluate tan
51. Use the properties of inverse trigonometric
˘
È
functions to evaluate tanÍÍÍÎ arctan(0.31) ˙˙˙˚ .
ÊÁ
ÁË
52. Find the exact value of csc ÁÁÁÁ arctan
11 ˆ˜˜˜
˜.
60 ˜¯
53. Write an algebraic expression that is equivalent
ÊÁ
x
to sinÁÁÁÁ arctan
6
Ë
54. Use a graphing utility to graph the function below.
ÊÁ x
y = −2 + arctanÁÁÁ
ÁË 2
ˆ˜
˜˜˜
˜
¯
55. Use a graphing utility to graph the function below.
−1
y = −2 tan (3x )
8
ˆ˜
˜˜˜ .
˜
¯
Name: ________________________
ID: A
56. If a = 6 and c = 19, determine the value of A.
59. A land developer wants to find the distance
Round to two decimal places.
across a small lake in the middle of his
proposed development. The bearing from A to
B is N 14°W. The developer leaves point A and
travels 53 yards perpendicular to AB to point
C. The bearing from C to point B is N 76°W.
Determine the distance, AB, across the small
lake. Round distance to nearest yard.
57. After leaving the runway, a plane's angle of
ascent is 15° and its speed is 265 feet per
second. How many minutes will it take for the
airplane to climb to a height of 11,000 feet?
Round answer to two decimal places.
58. A plane is 48 miles west and 49 miles north of
an airport. The pilot wants to fly directly to the
airport. What bearing should the pilot take?
Answer should be given in degrees and
minutes.
60. If the sides of a rectangular solid are as shown, and s = 6, determine the angle, θ, between the diagonal of
the base of the solid and the diagonal of the solid. Round answer to two decimal places.
9
ID: A
Ma1316x1R
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
ANS:
ANS:
ANS:
ANS:
ANS:
E
D
C
B
C
PTS:
PTS:
PTS:
PTS:
PTS:
1
1
1
1
1
OBJ:
OBJ:
OBJ:
OBJ:
OBJ:
Use trig identities to determine equivalency
Determine graph of trig function involving damping factor
Determine equivalent inverse trig functions
Use graphing utilities to graph inverse trig functions
Use graphing utilities to graph inverse trig functions
SHORT ANSWER
6. ANS:
3rd quadrant
PTS: 1
7. ANS:
OBJ: Determine in which quadrant an angle lies
PTS: 1
8. ANS:
OBJ: Determine two coterminal angles (degrees)
231°, − 129°
329° 28' 26"
PTS: 1
9. ANS:
OBJ: Convert decimal degree to DMS form
62°
PTS: 1
10. ANS:
OBJ: Find the supplement of an angle (degrees)
4th quadrant
PTS: 1
11. ANS:
OBJ: Determine in which quadrant an angle lies
PTS: 1
12. ANS:
OBJ: Determine two coterminal angles (radians)
PTS: 1
13. ANS:
OBJ: Find the complement of an angle (radians)
PTS: 1
OBJ: Convert radian measure to degree measure
11π
5π
,−
4
4
5π
14
−70°
1
ID: A
14. ANS:
−3.836
PTS: 1
15. ANS:
OBJ: Convert degree measure to radian measure
10
7
PTS: 1
16. ANS:
OBJ: Find measure of central angle given radius and arc length
PTS: 1
17. ANS:
OBJ: Find length of arc given radius and central angle
S = 11.00 meters
A=
11π 2
m
3
PTS: 1
18. ANS:
OBJ: Find the area of a sector given the radius and central angle
875.4 rpm
PTS: 1
19. ANS:
OBJ: Determine revolutions per minute
25
24
PTS: 1
20. ANS:
sin θ =
OBJ: Determine trig value from diagram
3
10
PTS: 1
21. ANS:
tan30° =
PTS: 1
22. ANS:
OBJ: Determine value of a trig function
3
3
OBJ: Determine trig value given sin and cos
2.6131
PTS: 1
23. ANS:
OBJ: Calculate a trigonometric value using a calculator
θ = 60°
PTS: 1
OBJ: Determine theta without the use of calculator
2
ID: A
24. ANS:
x=
7
tan34°
PTS: 1
25. ANS:
OBJ: Determine value of variable using right-triangle trig
23 feet
PTS: 1
26. ANS:
OBJ: Application: Right triangle trig
sin θ = −
24
25
PTS: 1
27. ANS:
secθ = −
OBJ: Determine trig value from diagram
13
5
PTS: 1
28. ANS:
OBJ: Determine value of trig function given point on terminal side
Quadrant I
PTS: 1
29. ANS:
cscθ =
OBJ: Determine quadrant given constraints
13
12
PTS: 1
30. ANS:
tanθ = −
PTS: 1
31. ANS:
OBJ: Determine exact value of trig function given constraints
7
24
OBJ: Determine exact value of trig function given constraints
1
PTS: 1
32. ANS:
OBJ: Determine exact value of quadrant angle
2
2
PTS: 1
33. ANS:
OBJ: Determine exact trig value of angle
–0.4663
PTS: 1
OBJ: Calculate the value of a trigonometric function using a calculator
3
ID: A
34. ANS:
θ=
2π 4π
,
3
3
PTS: 1
35. ANS:
cos t = −
OBJ: Solve trig equations
3
2
PTS: 1
36. ANS:
OBJ: Evaluate trig function using unit circle
–1.7935 cm
PTS: 1
37. ANS:
period:
OBJ: Values of trig functions at any angle
8π
; amplitude:4
3
PTS: 1
38. ANS:
OBJ: Determine period and amplitude of trig graph
The period of g(x) is seven times the period of f(x).
Graph of g(x) is shifted downward 11 unit(s) relative to the graph of f(x).
PTS: 1
39. ANS:
PTS: 1
OBJ: Determine translations of trig graph
OBJ: Sketch graphs of trig functions
4
ID: A
40. ANS:
PTS: 1
41. ANS:
a = 2; b = 2; c = −
PTS: 1
42. ANS:
y=
OBJ: Use graphing utilities to graph trig functions
π
2
OBJ: Determine a and d of a trig function from a graph
πx
1
csc
2
3
PTS: 1
43. ANS:
PTS: 1
OBJ: Determine function given graph
OBJ: Sketch graphs of trig functions
5
ID: A
44. ANS:
PTS: 1
45. ANS:
OBJ: Use graphing utilities to graph trig functions
PTS: 1
46. ANS:
OBJ: Use graphing utilities to graph trig functions
As x → 0, f(x) → ∞.
PTS: 1
47. ANS:
OBJ: Determine effect of damping factor as function approaches zero
π
6
PTS: 1
OBJ: Determine value of inverse trig function without a calculator
6
ID: A
48. ANS:
−
π
6
PTS: 1
49. ANS:
OBJ: Determine value of inverse trig function without a calculator
–0.72
PTS: 1
50. ANS:
θ = arcsin
PTS: 1
51. ANS:
OBJ: Evaluate inverse functions
x
3
OBJ: Rewrite theta as an inverse function involving x
0.31
PTS: 1
52. ANS:
OBJ: Evaluate inverse trig functions
61
11
PTS: 1
53. ANS:
OBJ: Find the exact value of an expression involving inverse function
x
2
x + 36
PTS: 1
54. ANS:
PTS: 1
OBJ: Rewrite inverse trig expression as an algebraic expression
OBJ: Use graphing utilities to graph inverse trig functions
7
ID: A
55. ANS:
PTS: 1
56. ANS:
OBJ: Use graphing utilities to graph inverse trig functions
18.41°
PTS: 1
57. ANS:
OBJ: Find a third triangle given two pieces
2.67 minutes
PTS: 1
58. ANS:
OBJ: Application: Angle of ascent
135° 35'
PTS: 1
59. ANS:
OBJ: Find bearings
28 yards
PTS: 1
60. ANS:
OBJ: Find distance using surveying bearings
24.09°
PTS: 1
OBJ: Angle inside a solid using Pythagorean theorem and arctan
8