Review of Trigonometry
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. The graph depicts the height in metres of a bouncing ball with respect to time in seconds (given the friction in
the environment to be negligible). What is the period of the graph?
height
3
2
1
1
2
3
4
5
6
7
8
9
10 time
a. 2 seconds
b. 10 seconds
c. 4 seconds
d. 5 seconds
____
2. Consider a Ferris wheel which loads passengers at a height of 1 metre above the ground a carries them to a
height of at most 15 metres. What is the amplitude of the function which models the height of a passenger
above ground while in constant motion?
a. 7
c. 1
b. 14
d. 15
____
3. What is the equation of the axis of the graph?
11
10
9
8
7
6
5
4
3
2
1
–1
a. y = 2
b. x = 2
____
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
y
1
x
c. y = 7
d. x = 7
4. The height of a passenger on a Ferris wheel with respect to time is given by a sinusoidal function. How would
you increase the amplitude of the function?
a. slow the rate of the Ferris wheel rotation
b. increase the height of the platform used to load passengers
c. increase the radius of Ferris wheel
d. increase the rate of the Ferris wheel rotation
Short Answer
5. A Ferris wheel loads passengers at a height 1 metre above the ground and carries them to height at most 15
metres above the ground. The wheel rotates at a rate of 1 revolution every 2 minutes. Draw the graph of an
approximate function which models the height of a passenger on the Ferris wheel with respect to time. Make
sure to include two periods.
6. What is the equation of the axis of the function?
y
1
2
3
4
5
6
7
8
9
10
x
–7
–8
–9
7. What is the amplitude of the graph? What is the equation of the axis?
y
14
13
12
11
10
9
8
7
6
5
–4
–3
–2
–1
1
x
8. Find the amplitude, period and equation of axis of the sinusoidal function.
y
7
6
5
4
3
2
1
1
x
Problem
9. What is the equation in the form y = asin k( + b) + d that best models the data in the graph below?
10. A radio transmission tower sways sinusoidally in high winds. The top of the tower moves 75 cm back and
forth from its normal vertical position. A motion detector measures that its top is 75 cm to the left of vertical
at t = 10 s and 75 cm to the right of vertical at t = 40 s. Determine the simplest cosine function that models
this motion.
11. The average monthly temperature for a location in Ontario as a function of month number can be modelled
using the equation y = acos k(+ b) + d. If the highest average monthly temperature is 20°C and the lowest
average monthly temperature is –10°C, which occurs in January (month 0) with an annually repeating pattern,
write the simplest equation that models this relationship and sketch its graph.
12. The depth of the water in an ocean harbour varies due to the tides and can be modelled using the equation y =
asin k( + b) + d. High tide occurs at 5:00 A.M. with a water depth of 16 m and low tide occurs at 5:00 P.M.
with a water depth of 5 m.
(a) Determine the equation that models this relationship if t = 0 at midnight.
(b) Sketch its graph.
(c) Determine the water depth at 2:00 P.M.
13. A ride at Canada's Wonderland called Jet Scream is like a large pendulum that swings back and forth
periodically with a constant amplitude during the middle part of the ride. During this part of the ride, the
passenger section makes six complete swings in one minute. The passengers are 2.5 m above the ground at
the lowest point and 6.0 m above the ground at the highest point. Let t = 0 at the highest point of the ride.
(a) Write the equation that models this motion in the form y = asin k( + b) + d.
(b) Calculate the height of the passengers at 1.4 min.
14. An electronics engineer uses an oscilloscope to analyze the voltage signal in an electronic circuit. The
oscilloscope produces a graph on its display screen as shown below. The vertical scale is in millivolts and the
horizontal scale is in milliseconds. Use the following formulas to find the RMS voltage, VRMS, and the
frequency, f, of the signal.
VRMS = 2  amplitude 
;
15. A bicycle wheel with diameter 0.5 metres is spinning at a constant rate of 2 revolutions per second. Draw an
approximate function which models the height of a point on the wheel above the ground with respect to time.
(include at least two cycles)
16. A boy on a Ferris wheel that turns at a constant rate of 1 revolution every 3 minutes is at most 23 metres
above the ground and at least 2 metres above the ground. Construct a graph that models the boy’s height
above ground with respect to time.
17. Sketch a graph of the function sin(x – 45) + 2.
18. Sketch a graph of the function sin(x + 30) – 3.
19. Sketch a graph of the function sin(x + 45) – 6.
20. Sketch a graph of the function f(x) = –2sin x.
Review of Trigonometry
Answer Section
MULTIPLE CHOICE
1. ANS: A
PTS: 1
REF:
2. ANS: A
PTS: 1
REF:
OBJ: 6.3 - Investigating the Sine Function
3. ANS: A
PTS: 1
REF:
OBJ: 6.3 - Investigating the Sine Function
4. ANS: C
PTS: 1
REF:
OBJ: 6.4 - Comparing Sinusoidal Functions
Application
Thinking
OBJ: 6.2 - Periodic Behaviour
Application
Thinking
SHORT ANSWER
5. ANS:
height
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
minutes
PTS: 1
REF: Thinking
6. ANS:
The equation of the axis is y = –8.
OBJ: 6.2 - Periodic Behaviour
PTS: 1
REF: Application OBJ: 6.3 - Investigating the Sine Function
7. ANS:
The amplitude is 4. The equation of the axis is y = 10.
PTS: 1
REF: Thinking
OBJ: 6.3 - Investigating the Sine Function
8. ANS:
The amplitude is 2, the period is 1 and the equation of the axis is y = 3.
PTS: 1
REF: Knowledge and Understanding
OBJ: 6.4 - Comparing Sinusoidal Functions
PROBLEM
9. ANS:
As seen in the graph, the amplitude is 4, so a = 4. The central axis is y = –1, so the graph is shifted down by –
1 and d = –1. The period is 180°, making k = 2. Finally, the graph is horizontally shifted 30° to the right, so b
= 30°. Therefore, the equation is
.
PTS: 1
DIF: U/C
REF: Knowledge and Understanding
OBJ: 5.7 Modelling Periodic Phenomena
STA: TF4.01
TOP: Modelling Periodic Functions
10. ANS:
From the information given, the amplitude is 75 cm. Since it takes 30 s to move from the far left position to
the far right position, the period is double this time or 60 s. Normally, the cosine is at its maximum value at t
= 0. In this case it is at 10 s, therefore, the phase shift is –10 s. In this model, the normal position is when the
displacement is at 0 cm, so there is no vertical shift in the equation (d = 0).
Therefore, a = 75,
, b = –10, and the equation is
PTS: 1
DIF: U/C
REF: Application OBJ: 5.7 Modelling Periodic Phenomena
STA: TF4.04
TOP: Modelling Periodic Functions
11. ANS:
Using the information given, we can calculate the required constants.
|a| = 20 – 5 = 15; however, it is negative because the graph starts at negative values.
b = 0 because the graph has its minimum value at time zero when the cosine function is used.
Therefore, the equation is y = –15cos 0.52 + 5.
PTS: 1
STA: TF4.04
12. ANS:
DIF: U/C
REF: Application
TOP: Modelling Periodic Functions
OBJ: 5.7 Modelling Periodic Phenomena
(a) Using the information given in the question, the constants may be determined.
The water would be at the middle depth 6 h before 5:00 A.M. or at –1 h, so b = 1.
(a) Therefore, the equation is d = 5.5sin 0.26(t + 1) + 10.5, where d is the depth in metres and t is the time in
hours.
(b)
(c)
2:00 P.M., t = 14, so d = 5.5sin 0.26(14 + 1) + 10.5
At
6.7 m
PTS: 1
DIF: U/C
REF: Thinking/Inquiry/PS
OBJ: 5.7 Modelling Periodic Phenomena
STA: TF4.04
TOP: Modelling Periodic Functions
13. ANS:
(a) Using the information given in the question, the constants may be determined.
To use the sine function to model this motion, the graph needs to be shifted by
left, so b = 2.5.
period or 2.5 s to the
Therefore, the equation is h = 1.75sin 0.63(t + 2.5) + 4.25, where h is the height in metres and t is the time
in seconds.
(b)
PTS: 1
DIF: U/C
REF: Thinking/Inquiry/PS
OBJ: 5.7 Modelling Periodic Phenomena
STA: TF4.04
TOP: Modelling Periodic Functions
14. ANS:
From the graph, distance between adjacent x-intercepts is 400 ms, so the period is 400 ms.
RMS voltage = VRMS
= 2  amplitude 
= 2  10 
14.14 mV
PTS: 1
STA: TF4.04
15. ANS:
DIF: U/C
REF: Application
TOP: Modelling Periodic Functions
OBJ: 5.7 Modelling Periodic Phenomena
height
0.5
time
If the point on the wheel begins in the downward position, it should increase to a height at most 0.5 metres
then decrease back to an initial height of 0 metres in 0.5 seconds. Included are 2 cycles for a total interval of 1
second.
PTS: 1
REF: Application OBJ: 6.2 - Periodic Behaviour
16. ANS:
The function is sinusoidal with amplitude 10.5 metres and period 3 minutes.
y
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
x
PTS: 1
REF: Thinking
OBJ: 6.3 - Investigating the Sine Function
17. ANS:
The graph should transform sin x by translating up 2 units and to the right 45.
y
4
3
2
1
90
180
270
360
x
PTS: 1
REF: Thinking
OBJ: 6.5 - Transformations of the Sine Function: f(x) = sin(x - c) and f(x) = sin x + d
18. ANS:
The graph should translate the function sin x to the left 30 and down 3 units.
y
90
180
270
360
x
–1
–2
–3
–4
PTS: 1
REF: Thinking
OBJ: 6.5 - Transformations of the Sine Function: f(x) = sin(x - c) and f(x) = sin x + d
19. ANS:
The graph should be a transformation of sin x translated to the left 45 and down 6.
y
90
180
270
360
x
–5
–6
–7
PTS: 1
REF: Application
OBJ: 6.5 - Transformations of the Sine Function: f(x) = sin(x - c) and f(x) = sin x + d
20. ANS:
The graph retains the same x-intercepts as sin x, but the amplitude is 2 and the function is reflected over the xaxis.
y
2
1
–270
–180
–90
90
180
270
x
–1
–2
PTS: 1
REF: Thinking
OBJ: 6.6 - More Transformations of sin x: f(x) = a sin x