Download Test-3-trig-functions-review

Name: ________________________ Class: ___________________ Date: __________
ID: A
Test 3 Trig Functions
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. What is the value of sec 70 to the nearest thousandth?
A. 0.342
B. 0.364
____
2. What is the value of cos (295) to the nearest thousandth?
A. 2.364
B. –0.423
____
C. –230º
D. 40º
6. Which of these angles is NOT coterminal with an angle of 240 in standard position?
A. 120º
B. –600º
____
C. 50º
D. –130º
5. Which of these angles is coterminal with an angle of 230 in standard position?
A. –130º
B. 130º
____
C. 0.052
D. –19.231
4. What is the measure of the reference angle for an angle of 310 in standard position?
A. 310º
B. –50º
____
C. 0.423
D. 2.145
3. What is the value of cos 453 to the nearest thousandth?
A. –19.081
B. –0.052
____
C. –2.924
D. 2.924
C. –60º
D. 480º
7. Which expression represents the measures of all the angles coterminal with 301 in standard position?
A. 301  k180, k  Z
B. 59  k360, k  Z
C. 301  k360, k  ò
D. 301  k360, k  Z
1
Name: ________________________
____
8. Given tan 
A. cot 
ID: A
8
, which statement is true for all possible values of ?
9
9
8
9
8
8
C. cot  
9
D. cot cannot be determined
B.
____
cot  
9. What is the length of the arc that subtends a central angle of 150 in the unit circle?
A.
5
 units
6
C.
B.
75 units
D.
6
 units
5
5
 units
12
____ 10. In the unit circle, the length of the arc that subtends a positive central angle of  is
1
 units.
3
What is the measure of  in degrees?
1

540
D. 60
A. 120
B.
C.
60
____ 11. In a circle with radius 8 units, the length of the arc that subtends a positive central angle of  is
20
 units. What is the measure of  in degrees?
9
A. 100
B. 50
C. 50
D. 400
____ 12. What is 85 in radians?
A. 85 radians
B.
15 300

radians
C.
D.
17
radians
36
17
 radians
36
____ 13. What is 240 in radians?
A. –240 radians
B.
4
 radians
3
C.
43 200

radians
4
D.   radians
3
2
Name: ________________________
ID: A
____ 14. What is 695 in radians?
A.
B.
139
 radians
36
125 100
radians
C. 695 radians
D.

139
radians
36
____ 15. What is –6 radians in degrees?
A. –344
B. –19
ÊÁ 4
____ 16. What is the value of sec ÁÁÁÁ 
Ë 7
A. 1.19
B. 1.00
C. –1080
D. –2
ˆ˜
˜˜ to the nearest hundredth?
˜˜
¯
C. –0.22
D. –4.49
____ 17. What is the value of csc (3.7) to the nearest hundredth?
A. 1.89
B. –15.50
C. 1.24
D. 0.53
____ 18. In a circle with radius 5 cm, what is the length of the arc that subtends a central angle of
3
radians? Give the answer to the nearest tenth.
4
A. 11.8 cm
B. 0.5 cm
C. 3.8 cm
D. 2.4 cm
2
____ 19. Which angle is NOT coterminal with an angle of   radians in standard position?
5
A. 
B.
12

5
8

5
C. 0
D.
3
18

5
Name: ________________________
ID: A
____ 20. Which function below has this graph?
A. y  sin x
B. y  tan x
C. y  cos x
D. None of the above
____ 21. Which function below has this graph?
A. y  sec x
B. y  csc x
C. y  cot x
D. None of the above
4
Name: ________________________
ID: A
____ 22. Which set of functions describes these graphs?
A.
y  cos x
y  2 cos x
y  4 cos x
B.
y  cos x
y  cos 2x
y  cos 4x
C.
y  cos x
ÊÁ
y  cos ÁÁÁÁ x 
Ë
ÊÁ
y  cos ÁÁÁÁ x 
Ë
____ 23. What is the amplitude of the function y  9 sin x ?
A. 9
B. 18
C. 9
D. –9
____ 24. What is the amplitude of this sinusoidal function?
A. 3
B. 2
C. 6
D. –3
5
D.

2

4
ˆ˜
˜˜
˜˜
¯
ˆ˜
˜˜
˜˜
¯
y  cos x
y  cos x  2
y  cos x  4
Name: ________________________
ID: A
____ 25. What is the period of the function y  tan 5x ?
A.
B.

5
5

C.
D.
2
5

5
____ 26. What is the period of this function?
y  f(x)
A.
B.

2

4
C.
D.

2
2

ÁÊ
5 ˜ˆ˜˜
____ 27. What is the phase shift of the function y  cos ÁÁÁÁ x 
?
6 ˜˜¯
Ë
A.
B.
5
6
17
6
C.
D.
7
6
5
6
____ 28. This graph is the image of y  cos x after a phase shift.
Which value below could represent the phase shift?
A.
B.
13
6
11
6
C.
D.
6

6

6
Name: ________________________
ID: A
____ 29. Which function below describes this graph?
A.
B.
y  sin (x  3 )
y  sin x  3
C.
D.
y  sin x
y  3sin x
C.
D.
y  sin x  2
y  2sin x
____ 30. Which function below describes this graph?
A.
B.
y  sin x
y  sin 2x
____ 31. Which number is NOT in the domain of y  tan 3x ?
7
A.  
6
B.

4
C.  
3
1
D.  
3
7
Name: ________________________
ID: A
____ 32. Identify the transformations that would be applied to the graph of y  sin x to get the graph of
ÊÁ 1 ˆ˜
y  sin ÁÁÁÁ x ˜˜˜˜  1 .
Ë2 ¯
A. A vertical stretch by a factor of 2, and then a translation of 1 unit down
1
B. A horizontal compression by a factor of , and then a translation of 1 unit down
2
C. A horizontal stretch by a factor of 2, and then a translation of 1 unit down
1
D. A horizontal stretch by a factor of 1, and then a translation of units right
2
ÊÁ
2 ˆ˜˜˜
 9?
____ 33. What is the amplitude of the graph of y  8 sin 2 ÁÁÁ x 
ÁË
5 ˜˜¯
A. 16
B. 2
C. –1
D. 8
ÊÁ
5 ˆ˜˜˜
 6?
____ 34. What is the equation of the centre line of the graph of the function y  2 cos 5 ÁÁÁ x 
ÁË
6 ˜˜¯
A.
y  –6
B.
y  –4
y  –12
5
D. y 
6
C.
ÊÁ
5 ˆ˜˜˜
 8?
____ 35. What is the period of the function y  2 sin 3 ÁÁÁ x 
ÁË
7 ˜˜¯
A.
B.
2
3
7
5
C.
D.
5
7
3
2
ÊÁ
 ˆ˜
____ 36. What is the range of the function y  2 cos 3 ÁÁÁ x  ˜˜˜˜  5 ?
ÁË
5¯
A. 7  y  3
B. 3  y  7
C. 2  y  8
D. 3  y  7
8
Name: ________________________
ID: A
____ 37. Which function below best describes this graph?
A.
y  sin
B.
y  sin
3


3
(x  2 )  5
C.
y  sin
(x  2 )  5
D.
y  sin
3


3
(x  2 )  5
(x  2 )  5
2
(x  4.5 )  23 models the height, y metres, of a seat on a Ferris wheel at
9
any time x minutes after the wheel begins to rotate. What is the diameter of the wheel?
____ 38. Suppose the function y  19cos
A. 19 m
B. 9 m
C. 38 m
D. 42 m
____ 39. Which of the following angles, in degrees, is coterminal with, but not equal to,
A. 36°
B. 216°
C. 306°
D. 396°
____ 40. Determine the equation of a circle with centre at the origin and radius 8.
A. x 2  y 2  8
C. x 2  y 2  16
B.
x 2  y 2  64
D. x 2  y 2 
9
8
1
 radians?
5
Name: ________________________
ID: A
____ 41. Which graph represents an angle in standard position with a measure of 285°?
C.
A.
B.
D.
____ 42. Determine the measure of the angle in standard position shown on the graph below. Round your answer to the
nearest tenth of a degree.
A. 291.8°
B. 201.8°
C. 111.8°
D. 21.8°
10
Name: ________________________
ID: A
____ 43. The coordinates of the point that lies at the intersection of the terminal arm and the unit circle at an angle of
33° are
A. (0.84, 0.65)
C. (0.84, 0.54)
B. (0.65, 0.54)
D. (0.54, 0.84)
____ 44. Identify the point on the unit circle corresponding to an angle of
3 1
, )
2 2
C. (
3
3
,
)
2
3
3
1
( ,
)
2 2
D. (
3 1
, )
3 2
A. (
B.

radians in standard position.
6
____ 45. Which point on the unit circle corresponds to tan  = 0?
A. (1,1)
C. (0,0)
D. (1,0)
B. (0,1)
____ 46. If the angle  is 1400° in standard position, it can be described as having made
8
8
A. 3 rotations
C. 3 rotations
9
9
7
7
B. 7 rotations
D. 7 rotations
9
9
11
Name: ________________________
ID: A
5
____ 47. Which graph represents the function y = 2cos( x), where x is in degrees?
3
A.
C.
B.
D.
____ 48. The graph of y  sin x can be obtained by translating the graph of y  cos x

A.
units to the right
C.  units to the right
4


units to the right
D.
units to the right
B.
2
3
____ 49. Give an equation for a transformed sine function with an amplitude of
to the right, and a vertical translation of 3 units down.
9
A. y = sin 4 (x  7 / 8 ) – 3
C. y =
7
9 È
˘
B. y = sin ÍÍÍÎ 4 (x  7 / 8 ) ˙˙˙˚  3
D. y =
7
9
1
7
, a period of , a phase shift of rad
7
2
8
9 ÈÍ
˘
sin Í 4 (x  7 / 8 ) ˙˙˙˚  3
7 ÍÎ
9
sin 4 (x  7 / 8 ) – 3
7
____ 50. Which of the following is not an asymptote of the function f ()  tan ?
9
C. x = 
A. x  
2
7
3
B. x = 
D. x = 
2
2
12
Name: ________________________
ID: A
Short Answer
1. Sketch the angle –40° in standard position, then identify the reference angle.
2. Determine the exact value of sin(120).
3. Determine the exact value of csc 405.
4. For the point P(2,4) on the terminal arm of an angle  in standard position, determine the exact value of
cot .
5. Sketch the angle
1
 radians in standard position.
2
6. The graph of y  sin x is shown below for 0  x  2.
Extend the graph for x  2 and for x  0 .
13
Name: ________________________
ID: A
7. The graph of y  cos x is shown below.
On the same grid, sketch the graph of y  4 cos x .
8. The graph of y  cos x is shown below.
On the same grid, sketch the graph of y  cos 2x .
y  cos x
9. The graph of y  cos x is shown below.
On the same grid, sketch the graph of y  cos x  2 .
y  cos x
14
Name: ________________________
ID: A
10. The graph of y  sin x is shown below.
ÊÁ
 ˆ˜
On the same grid, sketch the graph of y  sin ÁÁÁ x  ˜˜˜˜ .
ÁË
6¯
y  sin x
11. Write a general equation for the asymptotes of the graph of y  tan (4x) .
ÊÁ
 ˆ˜
12. Identify the following characteristics of the graph of y  3cos 2 ÁÁÁÁ x  ˜˜˜˜ shown below.
2¯
Ë
• amplitude
• phase shift
• minimum value
• period
• zeros
• maximum value
• equation of the centre line
• domain
• range
ÁÊ
 ˜ˆ
y  3cos 2 ÁÁÁÁ x  ˜˜˜˜
2¯
Ë
13. Write an equation for a sine function with amplitude 8, period
shift 

4
.
15
2
, equation of centre line y  9 , and phase
3
Name: ________________________
ID: A
14. Write an equation that represents the sine function graphed below.
15. A table fan has a mark on the tip of one blade.
The equation y  17cos (6 x)  28 represents the height
of the mark, y centimetres, above the table x seconds
after the fan is turned on. What is the height of the mark
above the table when it is closest to the table?
ÈÍ Ê ˆ ˘˙ 2 ÈÍ Ê ˆ ˘˙ 2
ÍÍ Á 5 ˜˜ ˙˙
Í Á ˜˙
˜ ˙˙  ÍÍÍÍ sin ÁÁÁ 5 ˜˜˜ ˙˙˙˙ .
16. Find the exact value of ÍÍÍ cos ÁÁÁ
ÍÍ ÁË 6 ˜˜¯ ˙˙˙
ÍÍ ÁË 6 ˜¯ ˙˙
˚
˚
Î
Î
Problem
1. P(3,5) is a terminal point of angle  in standard position.
Determine all possible measures of  in the domain 740    20.
Give the answers to the nearest degree.
2. Given cot   1, determine all possible measures of angle  in the domain 2    2 .
16
Name: ________________________
ID: A
3. Graph y  2sin x .
Identify the amplitude, period, general
expression for the zeros, domain of the
function, and range of the function.
4. Graph y  sin 4x .
Identify the amplitude, period, general
expression for the zeros, general equation for
the asymptotes, domain of the function, and
range of the function.
17
Name: ________________________
ID: A
ÁÊ
 ˜ˆ
5. The graph of y  sin x is shown below. On the same grid, sketch the graph of y  4 sin 2 ÁÁÁÁ x  ˜˜˜˜  1.
4¯
Ë
Describe these characteristics of this function: amplitude, period, phase shift, equation of the centre line,
domain, and range
y  sin x
ÊÁ
2 ˆ˜˜˜
6. Sketch the graph of y  4sin ÁÁÁ 4x 
 2.
ÁË
3 ˜˜¯
Describe these characteristics of the function: amplitude, period, phase shift, equation of the centre line,
domain, and range
18
ID: A
Test 3 Trig Functions
Answer Section
MULTIPLE CHOICE
1. ANS:
REF:
LOC:
2. ANS:
REF:
LOC:
3. ANS:
REF:
LOC:
4. ANS:
REF:
LOC:
KEY:
5. ANS:
REF:
LOC:
KEY:
6. ANS:
REF:
LOC:
KEY:
7. ANS:
REF:
LOC:
8. ANS:
REF:
LOC:
9. ANS:
REF:
TOP:
10. ANS:
REF:
TOP:
KEY:
11. ANS:
REF:
TOP:
KEY:
12. ANS:
LOC:
13. ANS:
LOC:
D
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
KEY: Procedural Knowledge
C
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
KEY: Procedural Knowledge
B
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
KEY: Procedural Knowledge
C
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T1
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
A
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T1
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
C
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T1
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
D
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T1
TOP: Trigonometry
KEY: Conceptual Understanding
A
PTS: 1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
KEY: Conceptual Understanding
A
PTS: 1
DIF: Easy
6.2 Angles in Standard Position and Arc Length
LOC: 12.T1
Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
D
PTS: 1
DIF: Moderate
6.2 Angles in Standard Position and Arc Length
LOC: 12.T1
Trigonometry
Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills
C
PTS: 1
DIF: Moderate
6.2 Angles in Standard Position and Arc Length
LOC: 12.T1
Trigonometry
Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills
D
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
KEY: Procedural Knowledge
D
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
KEY: Procedural Knowledge
1
ID: A
14. ANS:
LOC:
15. ANS:
LOC:
16. ANS:
LOC:
17. ANS:
LOC:
18. ANS:
LOC:
KEY:
19. ANS:
LOC:
KEY:
20. ANS:
REF:
TOP:
21. ANS:
REF:
TOP:
22. ANS:
LOC:
23. ANS:
LOC:
KEY:
24. ANS:
LOC:
KEY:
25. ANS:
LOC:
26. ANS:
LOC:
27. ANS:
LOC:
28. ANS:
LOC:
KEY:
29. ANS:
LOC:
30. ANS:
LOC:
31. ANS:
LOC:
KEY:
32. ANS:
REF:
TOP:
A
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
KEY: Procedural Knowledge
C
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
KEY: Procedural Knowledge
D
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T3
TOP: Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T3
TOP: Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding
C
PTS: 1
DIF: Moderate
REF: 6.3 Radian Measure
12.T1
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
B
PTS: 1
DIF: Easy
6.4 Graphing Trigonometric Functions
LOC: 12.T4
Trigonometry
KEY: Conceptual Understanding
A
PTS: 1
DIF: Difficult
6.4 Graphing Trigonometric Functions
LOC: 12.T4
Trigonometry
KEY: Conceptual Understanding | Problem-Solving Skills
A
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Conceptual Understanding
C
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding
A
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding
D
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Procedural Knowledge
D
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
D
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Conceptual Understanding
B
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
KEY: Conceptual Understanding
A
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
12.T4
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
C
PTS: 1
DIF: Easy
6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge
2
ID: A
33. ANS:
REF:
TOP:
34. ANS:
REF:
TOP:
35. ANS:
REF:
TOP:
36. ANS:
REF:
TOP:
37. ANS:
REF:
TOP:
38. ANS:
REF:
TOP:
39. ANS:
NAT:
40. ANS:
NAT:
41. ANS:
NAT:
42. ANS:
NAT:
43. ANS:
NAT:
KEY:
44. ANS:
NAT:
NOT:
45. ANS:
NAT:
KEY:
46. ANS:
NAT:
NOT:
47. ANS:
NAT:
KEY:
48. ANS:
NAT:
KEY:
49. ANS:
NAT:
KEY:
D
PTS: 1
DIF: Easy
6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Easy
6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge
A
PTS: 1
DIF: Moderate
6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge
D
PTS: 1
DIF: Moderate
6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
D
PTS: 1
DIF: Moderate
6.7 Applications of Sinusoidal Functions
LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
C
PTS: 1
DIF: Moderate
6.7 Applications of Sinusoidal Functions
LOC: 12.T4
Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
D
PTS: 1
DIF: Average
OBJ: Section 4.1
T1
TOP: Angles and Angle Measure
KEY: radians | degrees
B
PTS: 1
DIF: Easy
OBJ: Section 4.2
T2
TOP: Unit Circle
KEY: unit circle | unit circle equation
C
PTS: 1
DIF: Easy
OBJ: Section 4.1
T1
TOP: Angles and Angle Measure
KEY: degrees
D
PTS: 1
DIF: Easy
OBJ: Section 4.1
T1
TOP: Angles and Angle Measure
KEY: radians
C
PTS: 1
DIF: Average
OBJ: Section 4.3
T2
TOP: Trigonometric Ratios
trigonometric ratios | unit circle | terminal arm | angle
A
PTS: 1
DIF: Average
OBJ: Section 4.3
T2
TOP: Trigonometric Ratios
KEY: exact value | unit circle | radians
tan90 and tan270 do not include undefined
D
PTS: 1
DIF: Average
OBJ: Section 4.3
T2
TOP: Trigonometric Ratios
Unit Circle | exact value | tangent ratio
C
PTS: 1
DIF: Average
OBJ: Section 4.1
T1
TOP: Angles and Angle Measure
KEY: rotations | standard position
Mixed numbers
B
PTS: 1
DIF: Difficult
OBJ: Section 5.1
T4
TOP: Graphing Sine and Cosine Functions
graph | amplitude | period | sinusoidal function
D
PTS: 1
DIF: Easy
OBJ: Section 5.2
T4
TOP: Transformations of Sinusoidal Functions
translation | primary trigonometric function
A
PTS: 1
DIF: Difficult
OBJ: Section 5.2
T4
TOP: Transformations of Sinusoidal Functions
transformations | equation | properties | sinusoidal function
3
ID: A
50. ANS: A
NAT: T4
PTS: 1
DIF: Easy
TOP: The Tangent Function KEY:
OBJ: Section 5.3
asymptote | tangent function
SHORT ANSWER
1. ANS:
Reference angle: 40°
PTS:
REF:
LOC:
KEY:
2. ANS:
1
DIF: Easy
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T1
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding | Communication
sin (120)  
3
2
PTS: 1
DIF: Moderate
REF: 6.1 Trigonometric Ratios for Any Angle in Standard Position
LOC: 12.T3
TOP: Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
3. ANS:
csc 405  2
PTS:
REF:
LOC:
KEY:
4. ANS:
1
DIF: Moderate
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding
cot  
PTS:
REF:
LOC:
KEY:
1
2
1
DIF: Moderate
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
Procedural Knowledge | Conceptual Understanding
4
ID: A
5. ANS:
PTS: 1
DIF: Easy
REF: 6.3 Radian Measure
LOC: 12.T1
TOP: Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding | Communication
6. ANS:
PTS: 1
LOC: 12.T4
7. ANS:
DIF: Easy
REF: 6.4 Graphing Trigonometric Functions
TOP: Trigonometry
KEY: Procedural Knowledge
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Procedural Knowledge | Communication
5
ID: A
8. ANS:
y  cos 2x
y  cos x
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Procedural Knowledge | Communication
9. ANS:
y  cos x
y  cos x  2
PTS: 1
DIF: Easy
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Procedural Knowledge | Communication
6
ID: A
10. ANS:
ÊÁ
 ˆ˜
y  sin ÁÁÁ x  ˜˜˜˜
ÁË
6¯
y  sin x
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Procedural Knowledge | Communication
11. ANS:
Equations may vary. For example:
x  (2k  1)

8
,k Z
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
12. ANS:
• amplitude: 3
• period: 
• equation of the centre line: y  0
•
•
•
•
•
•
phase shift:

2
7
5
3
  3 5 7
,
,
, , ,
,
,
zeros: 
4
4
4
4 4 4
4
4
domain: 2  x  2
minimum value: –3
maximum value: 3
range: 3  y  3
PTS: 1
DIF: Easy
REF: 6.6 Combining Transformations of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Procedural Knowledge
13. ANS:
ÊÁ
 ˆ˜
y  8 sin 3 ÁÁÁÁ x  ˜˜˜˜ +9
4¯
Ë
PTS: 1
DIF: Easy
REF: 6.6 Combining Transformations of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
7
ID: A
14. ANS:
Students’ answers may vary. For example:
2
(x  1 )  2.
y  2 sin
5
PTS: 1
DIF: Moderate
REF: 6.7 Applications of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
15. ANS:
11 cm
PTS: 1
DIF: Moderate
REF: 6.7 Applications of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge
16. ANS:
ÈÍ Ê ˆ ˘˙ 2 ÈÍ Ê ˆ ˘˙ 2 ÊÁ
ˆ˜ 2 Ê ˆ 2
˜˜
ÍÍ ÁÁ 5 ˜˜ ˙˙
ÍÍ ÁÁ 5 ˜˜ ˙˙
ÁÁ
3
ÁÁ 1 ˜˜
˜
ÍÍ cos ÁÁ ˜˜ ˙˙  ÍÍ sin ÁÁ ˜˜ ˙˙  ÁÁ 
ÍÍ Á 6 ˜ ˙˙
ÍÍ Á 6 ˜ ˙˙
ÁÁ 2 ˜˜˜  ÁÁÁ 2 ˜˜˜
˜
ÍÎ Ë ¯ ˙˚
ÍÎ Ë ¯ ˙˚
Á
Ë ¯
¯
Ë

3 1

4 4

1
2
PTS: 1
DIF: Difficult
TOP: Trigonometric Ratios
OBJ: Section 4.3
NAT: T3
KEY: exact value | unit circle
8
ID: A
PROBLEM
1. ANS:
The terminal arm of angle  lies in Quadrant 1.
ÁÊ 5 ˜ˆ
The reference angle is: tan 1 ÁÁÁÁ ˜˜˜˜ 
Ö 59
Ë3¯
In Quadrant 1,  
Ö 59
Sketch a diagram.
The angles that are coterminal with 59 in the domain 740    20 are approximately:
59  360  301
301  360  661
Possible values of  are approximately: 661 and 301.
PTS:
REF:
LOC:
KEY:
1
DIF: Moderate
6.1 Trigonometric Ratios for Any Angle in Standard Position
12.T3
TOP: Trigonometry
Conceptual Understanding | Procedural Knowledge
9
ID: A
2. ANS:
tan    1
Since tan  is negative, the terminal arm of angle  lies in Quadrant 2 or Quadrant 4.
The reference angle is: tan 1 (1) 
4
3
4

7
In Quadrant 4:   2  , or
4
4
In Quadrant 2:    


4
, or
Sketch a diagram.
An angle that is coterminal with
5
3
 2 
4
4
An angle that is coterminal with

7
 2 
4
4
3
in the domain 2    2 is:
4
7
in the domain 2    2 is:
4
So, the possible measures of angle  are
3 5 7

,
,
, and
4
4
4
4
PTS: 1
DIF: Moderate
REF: 6.3 Radian Measure
LOC: 12.T3
TOP: Trigonometry
KEY: Procedural Knowledge | Conceptual Understanding
10
ID: A
3. ANS:
The graph of y  2sin x is the image after the graph of y  sin x has been stretched vertically by a factor of 2.
The amplitude is 2.
The period is 2.
The zeros are k  , k  Z.
The domain is x  ò .
The range is 2  y  2.
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge | Communication
11
ID: A
4. ANS:
The graph of y  sin 4x is the image after the graph of y  sin x has been
1
horizontally compressed by a factor of .
4
y  sin 4x
The amplitude is 1.
1
The period is .
2
k
, k  Z.
The zeros are
4
There are no asymptotes.
The domain is x  ò .
The range is 1  y  1.
PTS: 1
DIF: Moderate
REF: 6.5 Trigonometric Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge | Communication
12
ID: A
5. ANS:
ÊÁ
 ˆ˜
2 ÁÁÁ x  ˜˜˜˜  1 to y  a sin b (x  c )  d :
ÁË
4¯
a  4, so the graph of y  sin x is stretched vertically by a factor of 4, and the amplitude is 4.
1
2
b  2, so the graph is compressed horizontally by a factor of , and the period is
, or  .
2
2
Compare y  4 sin



c   , so the graph is translated
units left, and the phase shift is  .
4
4
4
d  1, so the graph is translated 1 unit up, and the centre line has equation: y  1
The domain is: x  ò
The maximum value of the function is: 1  4  5
The minimum value of the function is: 1  4  3
So, the range is: 3  y  5
ÊÁ
 ˆ˜
y  4 sin 2 ÁÁÁ x  ˜˜˜˜  1
ÁË
4¯
y  sin x
PTS: 1
DIF: Moderate
REF: 6.6 Combining Transformations of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge | Communication
13
ID: A
6. ANS:
ÊÁ
2
Write y  4sin ÁÁÁ 4x 
ÁË
3
ÊÁ
 ˆ˜
y  4sin 4 ÁÁÁÁ x  ˜˜˜˜  2
6¯
Ë
ˆ˜
˜˜  2 in the form y  a sin b (x  c )  d .
˜˜
¯
Compare to y  a sin b (x  c )  d :

a  4, b  4, c   , and d  2
6
The graph of y  sin x is stretched vertically by a factor of 4, compressed horizontally by a factor of
reflected in the x-axis, and then translated

6
1
,
4
units left and 2 units up.
ÁÊ
2 ˜ˆ˜˜
y  4sin ÁÁÁÁ 4x 
2
3 ˜˜¯
Ë
a  4, so the amplitude is 4.
2

b  4, so the period is
, or .
4
2


c   , so the phase shift is  .
6
6
d  2, so the equation of the centre line is y  2.
The domain is x  ò .
The maximum y-value is 4 units above the centre line and the minimum y-value is 4 units below the centre
line, so the range is 2  y  6.
PTS: 1
DIF: Difficult
REF: 6.6 Combining Transformations of Sinusoidal Functions
LOC: 12.T4
TOP: Trigonometry
KEY: Conceptual Understanding | Procedural Knowledge | Communication
14