Unformatted text preview: Principles of Math 12  Trigonometry I Practice Exam 2 www.math12.com Trigonometry I Practice Exam Use this sheet to record your answers 1. NR 2. 19. 29. 2. NR 3. 20. 30. NR 1. 11. 21. 31. 3. 12. 22. 32. 4. 13. 23. 5. 14. 24. 6. NR 4. NR 5. 7. 15. 25. 8. 16. 26. 9. 17. 27. 10. 18. 28. Copyright © Barry Mabillard, 2006
Principles of Math 12  Trigonometry I Practice Exam www.math12.com
3 www.math12.com Trigonometry I Practice Exam
The transformation g (θ ) = f ( 2θ ) − 2 is applied to the graph of f (θ ) = sin θ . 1. The range of the new graph is
A.
B.
C.
D. −3 ≤ y ≤ −1
−2 ≤ y ≤ 0
−3 ≤ θ ≤ −1
−2 ≤ θ ≤ 0 Use the following information to answer the next question.
A satellite is orbiting a small planet, as shown in the following diagram. 2. The height of the satellite above the surface of the planet is, to the nearest km,
A.
B.
C.
D. 162 km
3952 km
4326 km
5162 km Numerical Response
1. π⎞
⎛π
⎞
⎛
If the point ⎜ , −2 ⎟ lies on the graph of f (θ ) = a cos ⎜ θ − ⎟ − 4 , then the value
4⎠
⎝
⎝2
⎠
of a, to the nearest tenth, is _________. Principles of Math 12  Trigonometry I Practice Exam 4 www.math12.com Use the following information to answer the next question.
The equation of a trigonometric function is π⎞
⎛
f (θ ) = k sin ⎜ θ − ⎟ − 3, k > 0
3⎠
⎝ 3. The range of this function is
A. −3k ≤ f (θ ) ≤ 3k
B. −k ≤ f (θ ) ≤ k
C. −3 − k ≤ f (θ ) ≤ −3 + k
D. 3 − k ≤ f (θ ) ≤ 3 + k 4. π⎞
⎛
The graph of y = cos ⎜ θ + ⎟ is identical to the graph of
2⎠
⎝
A. y = − cos θ
B. y = − sin θ π⎞
⎛
C. y = cos ⎜ θ − ⎟
2⎠
⎝
D. y = sin θ
5. π⎞
⎛
The yintercept of the graph represented by f (θ ) = −3cos ⎜ kθ + ⎟ − b is
2⎠
⎝
A.  b
B. 3  b
3−b
C.
k
−3 − b
D.
k Principles of Math 12  Trigonometry I Practice Exam 5 www.math12.com Use the following information to answer the next two questions.
The partial graph of a trigonometric function is displayed below. 6. An equation that correctly represents this graph is
A.
B.
C.
D. 7. f (θ ) = −4sin (θ − 300 ) − 7 f (θ ) = −4 cos (θ − 600 ) − 7
f (θ ) = −4sin (θ + 600 ) − 7 f (θ ) = 4 cos (θ + 300 ) − 7 If the graph above is to be represented by a function in radian mode, rather than
degree mode, the parameter(s) which must be changed are
A.
B.
C.
D. a and d
b
c
b and c Principles of Math 12  Trigonometry I Practice Exam 6 www.math12.com Use the following information to answer the next question.
Two trigonometric functions, f ( x ) and g ( x ) , are graphed below 8. A statement that correctly describes the relationship between the graphs at
point A is
A. f ( x ) = g ( A) B. g ( m ) = f ( m ) = k
C. f ( k ) + g (k ) = 2m
D. g ( m) = f ( k ) = m 9. If cot θ = − 3
and cscθ < 0 , then the value of sin θ is
4 4
5
4
B.
5
3
C. −
5
3
D.
5
A. − Principles of Math 12  Trigonometry I Practice Exam 7 www.math12.com 10. If cos A = 30
, 0 < θ < 900 , and B = 600 + A , then the value of sec B is
2 A. 300
1
B.
900
C. 0
D. undefined Use the following information to answer the next question.
The graph of a trigonometric function f(x) is shown below Numerical Response
2. If the graph above is to be represented in the form f (θ ) = a sin [b(θ − c)] + d ,
then the value of b, to the nearest hundredth, is _________. Principles of Math 12  Trigonometry I Practice Exam 8 www.math12.com Numerical Response
3. 11. 3
and tan θ > 0 , then the value of sin 2 θ − cos 2 θ is,
5
to the nearest hundredth, ________.
If cos θ = − The correct statement regarding the graphs of f (θ ) = a sin bθ and
g (θ ) = k sin ⎡b (θ − c ) ⎤ is
⎣
⎦ A. both graphs have a period equal to b
B. the yintercept of g (θ ) is a  k units lower than the yintercept of f (θ ) .
C. the θ  intercepts of g (θ ) are c units to the right of the θ  intercepts of f (θ )
D. the yintercept of g (θ ) is k, and the yintercept of f (θ ) is a. 12. A graph that has the same yintercept as y = cos θ is
A. y = 3 cos θ
B. y = cos 3θ
C. y = cos (θ − 3)
D. y = cos θ + 3 Principles of Math 12  Trigonometry I Practice Exam 9 www.math12.com Use the following information to answer the next question.
The partial graph of a trigonometric function is shown below. The
⎛π
⎞
⎛ 3π
⎞
graph has a maximum value A ⎜ ,112 ⎟ , and a minimum value B ⎜ , 28 ⎟
⎝2
⎠
⎝2
⎠ 13. An equation that correctly represents the graph shown above is π⎞
⎛
A. y = 42cos ⎜ θ − ⎟ + 28
2⎠
⎝
B. y = 42cos (θ − π ) + 70
π⎞
⎛
C. y = 42cos ⎜ θ − ⎟ + 70
2⎠
⎝
3π ⎞
⎛
D. y = 42 cos ⎜ θ −
⎟ + 70
2⎠
⎝ Principles of Math 12  Trigonometry I Practice Exam 10 www.math12.com Use the following information to answer the next question.
A point is on a terminal arm in standard position, as shown below.
16π
The standard angle of the terminal arm is
9 14. The reference angle θ is
2π
9
B. 320˚
63
C. −
16π
5π
D.
18
A. Principles of Math 12  Trigonometry I Practice Exam 11 www.math12.com Use the following information to answer the next question.
A sidewalk encloses a pieshaped field, as illustrated below. Numerical Response
4. 15. The total length of the sidewalk, correct to the nearest metre, is __________. If cosθ =
A.
B.
C.
D. 4
3π
, and
< θ < 2π , the value of cotθ is equal to
5
2 3
5
4
3
3
−
5
4
−
3 Principles of Math 12  Trigonometry I Practice Exam 12 www.math12.com 16. ⎛π 2 ⎞
The graphs of f (θ ) = sin 2θ and g (θ ) = cos 2θ intersect at the points ⎜ ,
⎜8 2 ⎟
⎟
⎝
⎠
⎛ 5π − 2 ⎞
and ⎜
⎟
⎜ 8 , 2 ⎟ . If the amplitude of each graph is quadrupled, the new points
⎝
⎠
of intersection will be
A.
B.
C.
D. 17. ⎛π 2 ⎞
⎛ 5π − 2 ⎞
⎜,
⎟ and ⎜ ,
⎟
⎜8 8 ⎟
⎜8
8⎟
⎝
⎠
⎝
⎠
⎛π 2
⎞
⎛ 5π − 2
⎞
− 4⎟
+ 4 ⎟ and ⎜ ,
⎜,
⎜8
⎟
⎜8 2
⎟
2
⎝
⎠
⎝
⎠
⎛π
⎞
⎛ 5π
⎞
⎜ , 2 2 ⎟ and ⎜ , −2 2 ⎟
⎝8
⎠
⎝8
⎠
⎛π 2 ⎞
⎜,
⎜ 2 2 ⎟ and
⎟
⎝
⎠ ⎛ 5π − 2 ⎞
⎜,
⎟
⎜2
2⎟
⎝
⎠ The terminal arm of a rotation angle in standard position passes through the point
(8k, 6k). If k > 0, then the exact values of sinθ, cosθ, and tanθ are
55 4
A. − , , −
34 3
34 3
B. − , , −
55 4
433
C.
,− ,−
544
37
3
D. − , , −
10 10 4 18. ⎛ 13π ⎞
The exact value of −3 tan ⎜
⎟ is
⎝6⎠
A. 3
B. − 3 3
3
D. undefined
C. − Principles of Math 12  Trigonometry I Practice Exam 13 www.math12.com Use the following information to answer the next question.
The average wing span of a particular species of butterfly is 8 cm.
However, the wing span for new butterflies varies in a periodic
manner from year to year. An equation that models the wing span is
w(t ) = cos3 t − sin(t − 3) + 8 , where w(t) is the wing span in cm, and
t is the time in years. 19. A biologist monitors the butterflies over a 25 year period. The range of the wing
span is, to the nearest tenth,
A. 0 ≤ w ( t ) ≤ 16.0
B. 6.7 ≤ w ( t ) ≤ 9.3
C. 6.8 ≤ w ( t ) ≤ 9.2
D. 7.0 ≤ w ( t ) ≤ 9.0 Principles of Math 12  Trigonometry I Practice Exam 14 www.math12.com Use the following information to answer the next question.
A student uses technology to draw the graph of y = tan θ , as shown
below. 20. The asymptotes of this graph occur at
A. ± nπ
B. ±2nπ
C.
D. 21. π 2 π 2 ±n π
2 ± nπ All of the following are coterminal angles to 150˚ except
A. 930˚
17π
B.
6
23π
C.
6
D. 3.67 rad Principles of Math 12  Trigonometry I Practice Exam 15 www.math12.com Use the following information to answer the next six questions.
A Ferris Wheel at an amusement park has riders get on at position A, which
is 3 m above the ground. The highest point of the ride is 15 m above
the ground. The ride takes 40 seconds for one complete revolution. 22. A function of the form h ( t ) = a cos [b(t − c)] + d can be used to accurately model the height of a Ferris Wheel over time. An equation that correctly models
the Ferris Wheel shown above is
A. h(t ) = −6 cos 9t + 9
B. h(t ) = −6 cos 40π t + 9
C. h(t ) = −6 cos
D. h(t ) = −6 cos π
3 t +9 π 20 t +9 Principles of Math 12  Trigonometry I Practice Exam 16 www.math12.com 23. The time for a rider, who starts at position A, to travel to position B (a rotation
of 135°) is
A.
B.
C.
D. 12 s
13 s
14 s
15 s 24. If the ride makes three complete rotations, the total amount of time a rider on the
Ferris Wheel will spend above 13 m, rounded to the nearest second, is
A.
B.
C.
D. 11 s
15 s
25 s
32 s Numerical Response
6. The height of the rider 22 seconds after the ride begins is, to the
nearest tenth, ________. 25. If the Ferris Wheel rotates counterclockwise, instead of the original clockwise
motion, the new graph is best represented by
A. changing the sign of the leading coefficient.
B. applying the transformation y = f (t − 40)
C. applying the transformation y = f ( −t )
D. using a sine function instead of a cosine function, with no change to the
parameters. 26. The ride operator decides to speed up the ride. This will affect parameter
A.
B.
C.
D. a
b
c
d Principles of Math 12  Trigonometry I Practice Exam 17 www.math12.com 27. If f (θ ) = sin 4θ , where 0 ≤ θ < 3π , then the number of vertical asymptotes in the
1
graph of
is
f (θ )
A.
B.
C.
D. 8
9
12
13 Use the following information to answer the next question.
The pendulum of a grandfather clock swings back and forth with a periodic
motion that can be represented by a trigonometric function. At rest, the
pendulum is 20 cm above the base . The highest point of the swing is
26 cm above the base, and it takes two seconds for a complete swing
back and forth. 28. A cosine equation that models the height of the pendulum as a function of time, if
the pendulum is released from the highest point, is
A.
B.
C.
D. h(t ) = 6 cos π t + 23
h(t ) = 3cos π t + 20
h(t ) = 3cos 2π t + 20
h(t ) = 3cos π t + 23 Principles of Math 12  Trigonometry I Practice Exam 18 www.math12.com 29. The general solution to the equation 2 sin θ = 3 is
5π
± nπ
6
6
π
5π
B. θ = ± 2nπ ,
± 2nπ
6
6
π
4π
C. θ = ± 2nπ ,
± 2nπ
3
3
π
2π
D. θ = ± 2nπ ,
± 2nπ
3
3
A. θ = 30. π ± nπ , An appropriate window setting for the graph of y = 20.1sin 2π
(t  265) + 6.2 is
300 A. x: [0, 17000, 5000], y: [20, 30, 10]
B. x: [265, 0, 50], y: [0, 12.4, 1]
C. x: [0, 600, 100], y: [15, 30, 5]
D. x: [0, 2π, 31. π 2
, y: [20, 30, 5] The graph of g (θ ) = sin [3θ − π ] is equivalent to the graph of y = sin θ after a
A. horizontal shift of π units right, then a horizontal stretch by a factor of
B. horizontal stretch by a factor of 1
, then a horizontal shift of π units right.
3 C. horizontal stretch by a factor of 3, then a horizontal shift of
D. horizontal stretch by a factor of 1
.
3 π 3 units right. 1
π
units right.
then a horizontal shift of
3
3 Principles of Math 12  Trigonometry I Practice Exam 19 www.math12.com 32. The domain of f (θ ) = cot 4θ is
nπ
4
nπ
B. x ∈ R, x ≠ ±
2
C. x ∈ R, x ≠ ± nπ
D. x ∈ R
A. x ∈ R, x ≠ ± Principles of Math 12  Trigonometry I Practice Exam 20 www.math12.com Use the following information to answer the next question. The sunrise and sunset times for Yellowknife
(adjusted to remove the effects of daylight
savings time) are given below.
June 21, 2006 Dec. 21, 2006
Sunrise 2.57 (2:34 AM) 10.18 (10:11 AM)
Sunset 22.75 (10:45 PM) 15.00 (3:00 PM) A sinusoidal equation of the form
T ( x ) = a cos [b( x − c)] + d can be used to
graphically model the time of sunrise or
sunset throughout the year, where T ( x ) is
the time of day (using decimal time format),
and x is the day of the year. Written Response – 10%
• Determine an equation modeling the time of sunrise in Yellowknife. • 1. Determine an equation modeling the time of sunset in Yellowknife. Principles of Math 12  Trigonometry I Practice Exam 21 www.math12.com • Using technology, graph the functions representing sunrise and sunset times
in Yellowknife. • Mathematically describe the transformations required to change the graph
of f ( x ) = cos x to the graph representing the sunset time in Yellowknife. • Determine the number of days Yellowknife experiences a sunrise earlier than
4:00 AM. • Determine the number of hours of daylight in Yellowknife on February 15. Principles of Math 12  Trigonometry I Practice Exam 22 www.math12.com Use the following information to answer the next question. A mechanic changing a tire rolls a wheel along the ground towards the car. The radius
of the wheel is 42 cm, and the speed of the wheel as it rolls is 2 revolutions per second. The diagram below illustrates the vertical motion of a point on the tire over time.
It is possible to model the height of this point using a sinusoidal function of the
form h(t ) = −a sin [b(t − c)] + d Principles of Math 12  Trigonometry I Practice Exam 23 www.math12.com Written Response – 10%
• Determine the length of time required for one revolution of the tire. • 2. State the numerical value for each of the parameters a, b, c, & d.
Parameter
a
b
c
d Value • Write a function representing the motion of the point in the form
h(t ) = − a sin [b(t − c) ] + d • Write a formula that predicts the times when contact between the point
and ground occur. Use this formula to determine the time when the point
touches the ground for the fifth time. • A second wheel, with a radius of 39 cm, is rolled at the same speed of
2 rev/second. Compare the parameters a, b, c, & d for this wheel with the
original wheel. Principles of Math 12  Trigonometry I Practice Exam 24 www.math12.com Use the following information to answer the next question. 1
The graph of f (θ ) = csc θ is shown below:
2 Written Response – 10%
3. • Complete the following table:
a  value
b  value
Phase Shift
Vertical Displacement
Period
Domain
Range
xintercepts
yintercepts
Asymptotes
(general equation) Principles of Math 12  Trigonometry I Practice Exam 25 www.math12.com • 1
in the space below. Then, write a function g (θ ) that
f (θ )
represents the graph you drew in. • Explain how the location of the asymptotes in f (θ ) can be predicted from the Sketch the graph of graph of g (θ ) . • ⎛ 10π ⎞
Determine the exact value of f ⎜
⎟
⎝3⎠ You have now completed the examination. Please check over your answers
carefully before selfmarking. Good luck on your real exam!
Principles of Math 12  Trigonometry I Practice Exam 26 www.math12.com ...
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 Spring '09
 JACK
 Calculus, Trigonometry, Yellowknife

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