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5
Chapter
Sine Functions
Trigonometric Functions
•Describe key properties of periodic functions arising from real-world applications, given a numeric or graphical representation.
•Predict, by extrapolating, the future behaviour of a relationship
modelled using a numeric or graphical representation of a periodic
function.
•Make connections between the sine ratio and the sine function by
graphing the relationship between angles from 0º to 360º and the
corresponding sine ratios, defining this relationship as the function
f(x) = sin x, and explaining why the relationship is a function.
•Sketch the graph of f(x) = sin x for angle measures expressed in
degrees, and determine and describe its key properties.
•Make connections, through investigation with technology,
between changes in a real-world situation that can be
modelled using a periodic function and transformations of the
corresponding graph.
•Determine, through investigation using technology, the roles of
the parameters a, c, and d in functions in the form f(x) = a sin x,
f(x) = sin x + c, and f(x) = sin(x – d), and describe these roles in
terms of transformations on the graph of f(x) = sin x with angles
expressed in degrees.
•Sketch graphs of f(x) = a sin x, f(x) = sin x + c, and f(x) = sin(x – d)
by applying transformations to the graph of f(x) = sin x, and state
the domain and range of the transformed functions.
•Collect data that can be modelled as a sine function, through
investigation with and without technology, from primary sources,
using a variety of tools, or from secondary sources, and graph the
data.
•Identify periodic and sinusoidal functions, including those that
arise from real-world applications involving periodic phenomena,
given various representations, and explain any restrictions that
the context places on the domain and range.
•Pose problems based on applications involving a sine function, and
solve these and other such problems by using a given graph or a
graph generated with technology from a table of values or from
its equation.
228
In nature, many things follow a
cycle or regular pattern. Some examples are the rise and fall of
tides, the rising and setting of the
sun, and the phases of the moon.
These regular patterns can be
described as periodic behaviours
and modelled using a sine function.
In this chapter, you will learn how
to identify periodic functions, relate
circles, the sine ratio, and the sine
function, and connect sine functions
with real-world situations.
Vocabulary
amplitude
coterminal angles
cycle
initial arm
period
periodic function
phase shift
sine curve
sine function
standard position
terminal arm
terminal point
unit circle
229
Prerequisite Skills
Trigonometry
1. Use the sine, cosine, or tangent ratio to determine the length of the side indicated,
to the nearest unit.
a)
b)
8
c)
:
-Xb
E
'-•
p
)&•
f
G
&%bb
&*Xb
F
+'•
6
c
7
;
9
2. Determine the measure of angle x, to the nearest degree.
a)
b)
A
c)
)(bb
x
I
B
H
-b
*b
&*Xb
+,bb
x
@
?
J
A
x
@
&(Xb
3. Use the Pythagorean theorem to determine the length of each unknown side, to the nearest
tenth of a unit.
a)
b)
6
(#+Xb
7
P
c)
s
I
J
b
11.6 cm
)#,Xb
,#-bb
&'#(bb
10.5 cm
8
H
Q
p
R
Evaluate Trigonometric Expressions
4. Evaluate with a calculator, to four decimal places.
a) sin 50°
b) cos 15°
c) tan 45°
5. Evaluate each expression for the value indicated. Round to four decimal places.
a) sin u, for u  60°
b) 5 cos u, for u  37°
c) tan u, for u  30°
d) 10 sin(u  20°), for u  90°
e) 3 cos u  8, for u  45° f)​ ​ sin(2u  30°)  3, for u  10°
_1 2
Draw Angles
6. Use a protractor to draw each angle.
a) 30°
b) 45°
c) 90°
d) 120°
e) 150°
f) 180°
g) 200°
h) 270°
230 MHR Functions and Applications 11 • Chapter 5
Plot Ordered Pairs
7. a) Plot the ordered pairs (24, 23), (23, 0), (22, 3), (21, 0), (0, 23),
(1, 0), (2, 3), (3, 0), and (4, 23).
b) Describe the pattern made by the points.
8. What are the signs of the x- and y-coordinates of a point (x, y) in each of the four quadrants?
Transformations of Quadratics
9. Write an equation for the quadratic function that results from each transformation.
a) The graph of y  x2 is translated 4 units upward.
b) The graph of y  x2 is translated 5 units to the left.
c) The graph of y  x2 is stretched vertically by a factor of 3.
d) The graph of y  x2 is stretched vertically by a factor of 2 and then
translated 1 unit to the right and 6 units down.
_ 1
e) The graph of y  x2 is reflected in the x-axis, compressed vertically by a factor of ​ ​,
3
and then translated 2 units upward.
10. Use transformations to sketch a graph of each parabola. Label the vertex of each parabola.
a) y  x2  4
b) y  (x  3)2
c) y  2x2
d) y  x2
e) y  ​ ​ x2
f) y  3(x  1)2  5
_1 2
Domain and Range
11. Write the meaning of each domain and range statement.
a) domain  {x ∈ R | x > 2}
b) range  {y ∈ R | y  6}
c) domain  {x ∈ R}
d) range  {y ∈ R | 2  y  2}
12. Sketch a graph of each function. Then, write the domain and range for each function.
a) y  2x  3
b) y  x2
c) y  2(x  3)2  4
d) y  (x  6)2  1
Chapter Problem
Many careers in the medical field use an understanding of systems
that exhibit regular patterns, or periodic behaviours. These systems
exist in the human body and in the diagnostic tools used to evaluate
them. In this chapter, you will see how the actions of the heart, lungs,
arms, and medical imaging can be modelled using periodic functions.
Prerequisite Skills • MHR 231
5.1
Periodic Functions
The Bay of Fundy, which stretches
between the provinces of New Brunswick
and Nova Scotia, has the highest tides in
the world. Typically, high tide reaches
11 m to 15 m in height as
100 billion tonnes of seawater flow in
and out of the Bay of Fundy during one
tide cycle.
To learn more about tides, currents, and
water levels go to www.mcgrawhill.ca/
functionsapplications11 and follow the
links.
Investigate
How can you identify a graph that represents periodic behaviour?
The graph shows the predicted height of water at Hopewell Cape,
New Brunswick, during high and low tides for the first seven days
of May 2007, beginning at midnight on May 1.
1. Describe the graph.
2.The graph illustrates
periodic behaviour,
which is a regular,
predictable occurrence.
Why are tides
considered to exhibit
periodic behaviour?
10.0
Height (m)
h
7.5
5.0
2.5
0
3.Consider the portion
1
2
3
4
Time (days)
5
6
of the graph from midnight
on May 3 to midnight on May 4 (from t  2 to t  3).
a) Estimate:
i) the maximum value, or the height of high tide
ii) the minimum value, or the height of low tide
iii) the time between high tides
iv) the time between low tides
b) What would one complete pattern of the graph represent?
4.Give three other examples of periodic behaviour. Explain what
characteristics make them periodic.
232 MHR Functions and Applications 11 • Chapter 5
t
A function is a periodic function if it repeats a pattern, or cycle, at
regular intervals. The period of the function is the horizontal distance
from the beginning of one cycle to the end of that cycle. The amplitude
of a periodic function is half the difference between its maximum value
and its minimum value.
periodic function
•a function that has a
pattern of y-values that
repeats at regular intervals
of its domain
cycle
y
-
•one complete pattern of a
periodic function
dcZXnXaZ
bVm^bjbkVajZ
)
%
)
-
Ä)
&'
&+
'%
Vbea^ijYZ
period
b^c^bjbkVajZ
•the horizontal length of one
cycle of a periodic function
') x
eZg^dY
amplitude
Example 1
•half the distance between
the maximum and minimum
values of a periodic function
Identify the Graph of a Periodic Function
Determine whether or not each function is periodic. If it is, determine
the period, amplitude, domain, and range.
a) b)
y
2
–2
0
y
4
2
2
4
6
x
0
–2
4
8
x
12
–2
Solution
a) This graph is not periodic because the graph does not repeat its
y-values. The maximum and minimum values keep changing.
b) This graph repeats its y-values at regular
intervals. It is periodic. To find the period of
this function, locate points representing the
beginning and end of a cycle. The period is
the difference in the x-coordinates.
Period 5 6.
y
4
(2, 4)
(8, 4)
2
0
4
8
12
x
–2
The amplitude is half the difference
between the maximum and minimum
values. For this function, the maximum value is 4 and the
minimum value is 3.
4  (3)
amplitude 5 ​ ​
2
5 ​ 7 ​
2
__ _ 5 3.5
domain 5 {x ∈ R | 22  x  16} range 5 {y ∈ R | 23  y  4}
5.1 Periodic Functions • MHR 233
Example 2
Find Periodic Function Values
y  f(x) is a periodic function.
y
2
a) Find the value of f(0). Determine two
y = f(x)
other values of x that give the same
value of f.
–2
b) Find the value of f(0.5). Determine
1
0
–1
1
2
x
2
x
–1
two other values of x that give the
same value of f.
–2
c) Determine the domain and range of
the function.
Solution
a) From the graph, f(0)  2. Two other
y
2 (0, 2)
values of x that give the same value
of f are x  2 and x  2.
y = f(x)
1
(0.5, 0)
b) From the graph, f(0.5)  0. Two
other values of x that give the same
value of f are x  2.5 and x  1.5.
–2
–1
0
1
–1
c) domain 5 {x ∈ R}
–2
range5 {y ∈ R | 22  y  2}
Key Concepts
• A periodic function repeats its y-values at regular intervals.
• The period of the function is the length of one cycle, measured along the horizontal axis.
maximum value
 minimum value .​
• The amplitude of the function is calculated as ​ ______
2
y
amplitude
x
0
period
234 MHR Functions and Applications 11 • Chapter 5
Communicate Your Understanding
C1 A person is bouncing on a trampoline. The graph represents the
height of the person above the ground over time.
Height (m)
h
a) Explain why the function is periodic.
b) Describe how you would determine the period of the function.
c) Describe how you would determine the amplitude of the
4
2
0
4
function.
8
12
Time (s)
t
C2 A weight on a spring is bouncing up and down.
Describe the graph of its height relative to time.
Practise
2. Determine the period, amplitude, domain,
A
For help with questions 1 and 2, refer to
Example 1.
and range for each periodic function.
a)
y
2
1. Determine whether or not each graph is
periodic. Justify your decision.
a)
–2
y
0
2
4
x
6
–2
2
–2
0
2
x
b)
–2
y
4
b)
2
y
4
–4
–2
0
4
–4
c)
–4
0
2
4x
4
8
8x
c) y
–4
y
4
0
0
x
–4
4x
–8
–4
5.1 Periodic Functions • MHR 235
For help with question 3, refer to Example 2.
3. y  f(x) is a periodic function.
y
4
2
5. The electrical current in homes and offices
is called alternating current, or AC. The
graph shows the current, in amperes, over
three cycles. Explain why the current would
be considered alternating and describe the
periodic nature of the function.
c
2
4
6
8
0.4
Current (A)
0
x
–2
0
–0.4
a) Find the value of f (2), f (4), and f (7.5).
0.02 0.04 0.06 t
Time (s)
b) Find two other values of x that give the
same values of f as you found in part a).
B
6. A sound wave from a pure musical note
Connect and Apply
4. An electrocardiogram (ECG) is a graphical
representation of the voltage, in millivolts,
generated by the heart muscle during a
heartbeat. An ECG provides information
about the performance of the heart.
The graph shows the cycle of a normal
heartbeat of a person. Determine the
maximum value, the minimum value, the
amplitude, and the period of this function.
looks like the graph on the left. When
you play a CD or an MP3 file with the
sound turned up, an amplifier “clips” the
sound to protect the speakers from being
damaged. In this case, the graph resembles
the one on the right.
bVm^bjbdjieji
Voltage (mV)
v
2.0
Original Signal
Clipped Signal
1.0
0
0.5
1.0 1.5
Time (s)
2.0
t
Go to www.mcgrawhill.ca/
functionsapplications11 and follow the
links to learn more about an ECG and the
cardiac cycle.
236 MHR Functions and Applications 11 • Chapter 5
Which characteristic of the sound wave
has been “clipped” by the amplifier—the
period or amplitude? Explain.
7. The table shows the number of daylight
hours in Sudbury, Ontario, on the 21st of
each month for one year. Go to
www.mcgrawhill.ca/functionsapplications11
and follow the links to explore similar data
for other locations.
Explain your reasoning.
b) A person’s heartbeat slows down during
periods of sleep and speeds up during
physical exercise. How would the graphs
from these two periods differ from each
other? How would they be similar?
Daylight (h)
Jan
9.2
Feb
10.7
Mar
12.2
Apr
13.9
May
15.2
Jun
15.8
Jul
15.3
Aug
13.9
Sep
12.3
Oct
10.7
Nov
9.2
Dec
8.6
9. This graph shows the time of day of the
sunrise and the sunset in Toronto, Ontario,
over a 1-year period.
t
24:00
Time of Day
Month
a) Does this represent a periodic function?
sunset
16:00
sunrise
08:00
00:00
J
F M A M J
J A S O N Dd
Date
a) Describe how the graphs would look if
a) Make a scatter plot of the data and draw
a curve of best fit.
b) Does this represent a periodic function?
Explain your reasoning.
c) What are the maximum and minimum
number of hours of daylight? When do
they occur?
d) Why do you think data for the 21st day
of each month is chosen?
8. Chapter Problem The graph shows the
volume of blood in the left ventricle of a
person’s heart over a time span of 5 s.
V
extended to the previous and following
years. Explain.
b) Describe what happened in early March
and early November. Would these
be periodic phenomena? Justify your
reasoning.
10. The graph shows the mean monthly high
and low temperatures, in degrees Celsius,
for Brockville, Ontario, using recorded data
from over 30 years.
y
High
Low
30
20
10
Volume (mL)
&'%
0
-%
2
4
6
8
10
12
x
–10
)%
%
&
'
(
Time (s)
)
t
a) Predict the high and low temperatures
for a day in April 2015.
b) How accurate would your predictions
be? Explain your reasoning.
5.1 Periodic Functions • MHR 237
11. The Spinning Wheel ride at a local fair is
a large vertical wheel that rotates around a
stationary axis. The table shows the height,
in metres, above the ground of a person
riding the Spinning Wheel.
Time (s)
Height (m)
0
1.5
2
4.2
4
6.8
6
4.5
8
2.3
10
1.5
12
4.2
14
6.8
16
4.5
18
2.3
20
1.5
22
4.2
24
6.8
a) Make a scatter plot of the data.
b) Do the data appear to be periodic?
Explain.
c) Draw a curve of best fit and extend it for
an entire ride that lasts 1 min.
d) What is the difference between the
maximum and the minimum height
reached during the ride?
Achievement Check
12. A child is swinging on
a swing in the park. The
width of her swing path is
4 m and one swing back
and forth takes about 2 s.
a) Draw a graph that
)b
represents the distance
from rest after t seconds.
b) Explain why the swing’s motion is a
periodic function.
c) When might a child’s swinging not be
periodic?
d) Other playground equipment also have
a periodic component. Explain how a
periodic function relates to a
teeter-totter, a merry-go-round, or a set
of monkey bars.
Extend
C
13. Basil is on a large vertical spinning wheel
at an amusement park. He starts his ride at
the lowest point, which is 3 m above the
ground. The radius of the Ferris wheel is
7 m, and it is rotating at 2.5 revolutions per
minute. Sketch a graph representing Basil’s
height above the ground, relative to time,
for a 2-min ride.
Career Connection
Bettina completed a three-year degree
of salt, and how light and sound travel
in earth and atmospheric sciences at
in wave patterns. This information
an Ontario university and then went
can be used to build a computer model
on to achieve her master’s degree in
of an ocean climate, which helps
oceanography. In her career as a physical
meteorologists to predict the weather. oceanographer, Bettina studies currents,
Bettina’s data also can be used to find
tides, and waves, and how they interact
out how sea levels have changed over
with the atmosphere. She collects,
time and if these changes might pose analyses, and interprets data on the
an environmental hazard.
temperature of the water, the amount
238 MHR Functions and Applications 11 • Chapter 5
5.2
Circles and the Sine Ratio
The Ferris wheel is named after George Washington Gale
Ferris, Jr., who designed an 80-m wheel for The Chicago’s
World Fair in 1893. This first wheel could carry
2160 people at a time in its 36 cars.
As you ride on a rotating Ferris wheel, your height above
the ground depends on the rotational angle. How can you
determine which rotational angles will return you to the
same location as where you started and how high you are
above the ground?
C^^[b
Investigate
•protractor
How are rotational angles that result in the same position
connected?
1.In a basketball slam-dunk competition, Chris performed a 360.
In a snowboard competition, Jennifer performed a 720. What
does each of these statements mean?
2.Draw each pair of angles on the
y
same set of axes, beginning with the
positive x-axis.
a) 20° and 380°
b) 112° and 472°
20°
0
x
c) 180° and 540°
d) 40° and 320°
3.How are the two angles in each part of step 2 related?
4.Calculate and compare each pair of sine values. What do you
notice? Explain why this happens.
a) sin 20° and sin 380°
b) sin 112° and sin 472°
c) sin 180° and sin 540°
d) sin 40° and sin 320°
5.Find three other pairs of angles that have the same characteristic
as those in step 4.
5.2 Circles and the Sine Ratio • MHR 239
standard position
•the position of an angle
when its vertex is at the
origin and its initial arm is
on the positive x-axis
An angle in standard position has its vertex at the origin with its initial
arm on the positive x-axis, and its terminal arm rotates about the origin.
If the rotation is counterclockwise, the angle is positive. If the rotation is
clockwise, the angle is negative.
Positive Angle
y
initial arm
•the ray of an angle in
standard position that is on
the positive x-axis
terminal
arm
x
Simple periodic motion can be observed while rotating through a circle.
OA represents the initial arm of an angle and OP represents the terminal
arm. If the lengths of the arms of the angle equal the radius, r, then point
A(r, 0) is the initial point of an angle. The terminal arm intersects the
circle at point P(x, y), the terminal point. The circle has equation
x2  y2  r2.
y
P(x, y)
r
�
0 x
coterminal angles
•angles in standard position
that share the same
terminal arm
x
–60°
terminal
arm
0
initial
arm
•the ray of an angle in
standard position that
rotates about the origin
•the point on a circle that
results from forming a
given angle in standard
position
initial
arm
60°
0
terminal arm
terminal point
Negative Angle
y
y
A(r, 0)
x
As point P rotates around the circle, it will repeat its position by adding
or subtracting multiples of 360°. These resulting repeated angles are
known as coterminal angles. If 2  1  360°n, where n is an integer,
then 1 and 2 are coterminal angles, and sin 1 and sin 2 will have the
same value. For example, if 1  30° and n  1, then 2  30°  360°(1),
or 390°.
y
y
r
0
Ex
240 MHR Functions and Applications 11 • Chapter 5
30°
r
P(x, y)
x
0
390°
P(x, y)
x
Find Coterminal Angles
Example 1
Find two coterminal angles for each of the given angles.
a) 75°
b) 120°
Solution
a) One coterminal angle for 75° is 75° plus one rotation of 360° in the
counterclockwise direction.
y
75°  360°(1)  435°
435°
75°
Another coterminal angle is 75° plus two
rotations of 360° in the counterclockwise
direction.
0
x
75°  360°(2)  795°
795°
b) One coterminal angle for 120° is
y
120° plus one rotation of 360° in the
counterclockwise direction.
120°
120°  360°(1)  480°
480°
Another coterminal angle is 120° plus one
rotation of 360° in the clockwise direction.
–240°
0
x
120°  360°(1)  240°
Find an Angle in Standard Position
Example 2
Find the radius of the circle and the measure of the angle in standard
position for each terminal point, P.
a) P(8, 6)
b) P(5, 12)
c) P(2, 5)
Solution
<PZX]V2^]]TRcX^]b
a) Sketch the angle in standard
y
position. Make a triangle by
drawing a perpendicular from
(8, 6) to the x-axis.
r2

x2

4
y2
r2  82  62
–8
–4
0
–4
r2  100____
–8
r  ±10
r
y
� x
r2  64  36
r  ±​100 ​ P(8, 6)
8
4
8
x
Notice the similarity
between the equation of
a circle centred at the
origin and the Pythagorean
theorem. This is because
every point on the circle
is the same distance from
the centre of the circle.
The radius of the circle is
the hypotenuse of a right
triangle. So, the square of
the radius will equal the
sum of the squares of the
x- and y-coordinates of any
point on the circle.
5.2 Circles and the Sine Ratio • MHR 241
Since the length of a radius is always positive,
10 must be rejected.
r  10
Since all three sides of the triangle are known, you can use any of
the primary trigonometric ratios to find . Use the sine ratio.
y
sin   ​ _
r ​ 6 ​ sin   ​ _
10
  36.9°
The measure of the angle in standard position is approximately
36.9°.
b) Sketch the angle in
standard position.
r2  x2  y2
y
P(–5, 12)
12
r2  (5)2  (12)2
y
r2  25  144
r2  169
____
r  ±​169 ​ r  ±13
Since the length of a
radius is always positive,
13 must be rejected.
r
8
4
�
x
–12
–8
–4
0
4
8
12
–4
–8
–12
r  13
Use the sine ratio to find .
y
sin   ​ _
r ​ 12 ​ sin   ​ _
13
  67.4°
A scientific or graphing calculator is
programmed to give an angle result from
90° to 90° for the inverse sine function,
sin1 . The calculator result is an angle
of approximately 67.4°. However, in this
case, the angle in standard position is in
the second quadrant. So, to find the correct
measure of , you must subtract the angle measure from 180°.
180°  67.4°  112.6°
The measure of the angle in standard position is
approximately 112.6°.
242 MHR Functions and Applications 11 • Chapter 5
x
c) Sketch the angle in standard
y
position.
4
r2  x2  y2
2
r2  (2)2  (5)2
r2  4  25
–4
r2  29
�
x
–2
y
___
r  ±​29 ​ 0
2
4
x
–2
r
–4
Since the length
___ of a radius is always
positive, ​ 29 ​ must be rejected.
___
r
​ 29 ​ P(–2, –5)
Use the sine ratio to find .
y
sin   ​ _
r ​ 5
___ ​ sin   ​ _
​ 29 ​ 
  68.2°
The calculator result is an angle of
approximately 68.2°. In this case the angle in standard position
is in the third quadrant. To find the correct measure of , subtract
the angle measure from 180°.
180°  (68.2°)  248.2°
The measure of the angle in standard position is
approximately 248.2°.
Example 3
Find the Coordinates of a Terminal Point
A terminal point, P(x, y), is on the unit circle such that it forms a
120° angle in standard position.
y
P(x, y)
unit circle
•a circle of radius 1 unit
that is centred at the
origin
1
120°
–1
0
1
x
–1
a) Find the coordinates of point P.
b) How do the coordinates of a point on the unit circle relate to
trigonometric ratios?
5.2 Circles and the Sine Ratio • MHR 243
Solution
The circle has equation x2  y 2  1, so r  1. Use the sine and
cosine ratios to find the coordinates of point P.
y
sin   ​ _
r ​ x ​ cos   ​ _
r
y
sin 120°  ​ _ ​ 1
sin 120°  y
x ​ cos 120°  ​ _
1
cos 120°  x
y  0.866
x  0.5
The coordinates of P are approximately (0.5, 0.866).
b) The ordered pair of any point on
the unit circle is the cosine and
sine of a given angle in standard
position.
y
x ​ sin  ​ _
cos   ​ _
r ​ r
y
x ​  ​ _ ​  ​ _
1
1
y
x
y
P(x, y) =
(cos �, sin �)
1
�
0
–1
1
–1
Since the equation of the unit circle
is x2  y2  1, then
(cos )2  (sin )2  1.
Key Concepts
• An angle in standard position has its vertex at the origin and its initial
y
arm on the positive x-axis. If the rotation of the terminal arm is
counterclockwise, the angle is positive. If the rotation of the terminal
arm is clockwise, the angle is negative.
• Coterminal angles are angles in standard position that share the
terminal
arm
�
0
x
same terminal arm. They can be generated by adding or subtracting
multiples of 360° to the original angle, or by using 2  1  360°n,
where n is an integer.
• A unit circle has radius 1 unit and is centred at the origin. It has equation x2  y2  1.
• Ordered pairs on the unit circle can be related to trigonometry with (x, y)  (cos , sin ) and
(cos )2  (sin )2  1.
• When using the sin1  function to find the measure of an angle in standard position that
lies in the second or third quadrant, subtract the resulting calculator angle from 180° to
obtain the correct angle measure.
244 MHR Functions and Applications 11 • Chapter 5
x
Communicate Your Understanding
C1 Copy and complete the table showing the sign of each of the trigonometric ratios as  rotates
around the unit circle.
First Quadrant
y
1
Second Quadrant
y
P(x, y) 1
P(x, y)
0
1x
–1
–1
0
–1
Fourth Quadrant
y
1
y
1
�
�
�
�
–1
Third Quadrant
1x
–1
0
1x
–1
P(x, y) –1
0
1x
–1
P(x, y)
Sign of cos 
Sign of sin 
Sign of tan 
C2 Use coterminal angles to help explain why the values of sin  are periodic.
Practise
For help with question 3, refer to Example 2.
A
For help with questions 1 and 2, refer to
Example 1.
1. Find two coterminal angles for each
given angle. Draw each set of coterminal
angles on the same set of axes in standard
position.
a) 90°
b) 45°
c) 83°
d) 0°
e) 130°
f) 180°
g) 205°
h) 294°
i) 310°
2. Which pairs of angles are coterminal?
Justify your answer.
3. An angle is in standard position and
has terminal point P. For each set of
coordinates for P, find the radius of the
circle in exact form and the measure of
the angle to the nearest tenth of a degree.
Include a diagram illustrating each angle.
a) P(5, 12)
b) P(4, 3)
c) P(6, 9)
d) P(7, 2)
For help with questions 4 to 7, refer to Example 3.
4. Find the coordinates of a terminal point,
c) 100° and 820°
P(x, y), for each angle in standard position
on the unit circle. Round to one decimal
place. Draw a diagram to illustrate each
angle.
d) 380° and 680°
a) 64°
b) 90°
c) 150°
e) 40° and 320°
d) 300°
e) 240°
f) 180°
a) 30° and 210°
b) 70° and 430°
f) 50° and 400°
5.2 Circles and the Sine Ratio • MHR 245
5. A terminal point, P(x, y), on the unit circle
forms the given angle in standard position.
Find the coordinates of point P. Round to
three decimal places.
10. Find the measure of angle . Find a
a) 45°
b) 120°
c) 150°
coterminal angle that is positive. Then,
find one non-coterminal angle, if possible, that gives the same value of sin . Round
your answers to the nearest degree.
d) 200°
e) 255°
f) 300°
a) sin   0.8660
6. Verify that (cos )2  (sin )2  1 for each
b) sin   0.7071
of the following angles.
c) sin   0.5
a)   50°
d) sin   0.2588
b)   160°
e) sin   0.9848
c)   250°
f) sin   0.8910
d)   780°
g) sin   1
7. Evaluate. Round to three decimal places.
11. The roller on a computer printer rotates at
Explain why the sign of the result is either
positive or negative.
2000 revolutions per minute. How many
degrees does it rotate
a) sin 20°
b) cos 138°
a) in 1 min?
c) tan 170°
d) sin 129°
b) in 20 s?
e) cos 90° f) tan 80°
Connect and Apply
B
8. Even though 30° and 150° are
non-coterminal angles, a calculator will
verify that sin 30°  sin 150°. With the
help of a diagram, explain why this is true.
9. Find the measure of angle . Then, find one
coterminal angle with the same value of
sin . Round your answers to the nearest
degree.
a) sin   0.8660
b) sin   0.7071
c) sin   0.5
d) sin   0.2588
e) sin   0.9848
f) sin   0.8910
g) sin   0
246 MHR Functions and Applications 11 • Chapter 5
c) in 45 s?
12. A rotating spotlight shines on a door after
rotating 128° from its starting position.
Through what angle has the spotlight
rotated when it has shone on the door for
the fifth time?
13. Determine the measure of angle  in
17. A robotic arm is 1.5 m long. Beginning
from the horizontal, it rotates through a
given angle. Find the distance from the
end of the arm to its starting position.
standard position, correct to one decimal
place.
a) sin   0.8660, if  is in the second
quadrant
b) sin   0.7502, if  is in the third
quadrant
c) tan   0.3345, if  is in the fourth
quadrant
d) cos   0.9122, if  is in the third
quadrant
e) cos  0.4550, if  is in the fourth
quadrant
f) tan   1.4521, if  is in the second
quadrant
14. Given angle  in standard position with
terminal arm in the second quadrant and
12
_
sin   ​ ​ , determine the value of cos .
13
15. A point on the unit circle has coordinates
(0.6157, 0.7880). What angle in
standard position would this point
reference?
16. The minute hand of a watch is 1 cm in
length and is pointing at the number 10 on
the watch face.
a) What angle does it make with the
number 3?
b) If the centre of the watch face is
plotted on a set of axes at the origin,
what ordered pair would represent the
number 10 on the watch face?
a)   60°
b)   240°
c)   300°
Extend
C
18. Given that angle  is in standard position
with terminal arm in the third quadrant
_3
and cos   ​ ​ , determine the value of
5
cos   sin   tan .
19. Determine the area of the sector of the
circle shown.
y
1
(5, 0)
0
2
4
x
–2
(–3, –4)
–4
5.2 Circles and the Sine Ratio • MHR 247
Investigate the Sine Function
The Global Positioning
System (GPS) consists
of at least 24 GPS
satellites orbiting
Earth. These satellites
transmit signals
allowing GPS receivers
to determine the
receiver’s location,
speed, and direction.
90°
Latitude
5.3
0°
90°
–180°
–120°
–60°
0°
60°
120°
180°
Longitude
Orbiting at an altitude
of approximately 20 200 km, each GPS satellite repeats the same track
and configuration over any point approximately every 24 h. The path one
of these satellites takes as it orbits Earth is often shown on graphs as a
wave. This is because a three-dimensional path is being displayed on a
two-dimensional piece of paper.
Investigate A
C^^[b
•clear tape
•elastic band
•marker
•piece of paper 14.0 cm by
21.5 cm
•tennis ball
How can you generate a graph of the path of a satellite?
In this investigation, you will simulate the path of a satellite
as it orbits Earth.
1.Draw a line lengthwise along the middle of a piece of paper.
2.Tape the piece of paper around
the ball in a cylindrical shape,
so that the line on the paper
represents the equator of the ball.
3.Place an elastic band around the
paper-covered ball inclined at
about 30° to the equator. This
elastic band represents the path
of a satellite orbiting Earth. Use a
marker to draw a line around the
circumference of the paper-covered
ball by following the elastic band.
4.Unwrap the paper.
5.Describe the path of the curve
relative to the equator.
248 MHR Functions and Applications 11 • Chapter 5
6. This curve simulates the path of a satellite orbiting Earth,
translated to a two-dimensional graph showing longitudes from
0° to 360°. Draw a scale in increments of 30° along the line
representing the equator. Describe the path of the curve as the
longitude reaches 30°, 60°, 90°, 120°, and so on.
7.Does this curve represent a periodic function? Explain.
8.Explain why the displacement from the equator to the satellite’s
latitude is related to the sine of the degree longitude.
9.Repeat steps 1 through 4 using a steeper angle. What happens to
the curve? Why?
;XcTaPRh2^]]TRcX^]b
Displacement is the
directed distance from a
fixed position. It can be
positive or negative.
Investigate B
sine curve
How can you generate a sine curve?
•the graph of y = sin x
Method 1: Use Pencil and Paper
Work with a partner.
1. Use a paper plate to draw a circle on a large sheet of grid paper.
Draw a horizontal diameter. Measure the radius in centimetres,
accurate to the nearest millimetre.
2. Mark an angle at 15° by rotating
counterclockwise from the right side of the
circle. Measure the perpendicular height
from the diameter, in centimetres, of the
point where the angle intersects the circle.
15°
C^^[b
•compasses
•markers
•paper plate
•protractor
•ruler
•two large sheets of grid
paper
3. Repeat step 2, rotating counterclockwise in
steps of 15°, until you reach 360°. Record all your results in a
table with the headings shown. Consider what you learned in
Section 5.2, Circles and the Sine Ratio. What happens to the
height and the height  radius for angles greater than 180°?
Angle, x
Height (cm)
Height ÷ Radius, y
4.On a separate sheet of grid paper, label the x-axis with angles
from 0° to 720°, in 15° increments. Label the y-axis from 1 to 1,
so the distance from 1 to 1 equals the diameter of your circle.
Plot the ordered pairs (x, y) from your table and join them with a
smooth curve.
5.3 Investigate the Sine Function • MHR 249
5. The equation defining this graph is y  sin x. Explain why
sine would be the appropriate trigonometric ratio to use when
modelling the data.
sine function
•the function y = sin x,
which represents points on
the unit circle at angle x to
the x-axis
y
1
sin x
a) the amplitude
b) the period
c) the x-intercepts
d) the y-intercepts
P
e) the domain
x
–1
6.Describe each of the following for a sine function:
0
–1
1 x
f) the range
g)the intervals in which the function is increasing
and decreasing
7.Extend your graph so it continues to x  720°.
Explain why you extended it the way you did.
C^^[b
•graphing calculator
Method 2: Use a Graphing Calculator
1.Clear all functions from the Y= editor and ensure all
stats plots are turned off.
2. Make sure your calculator is in Degree
mode.
• Press
MODE
.
• Cursor down to the third line and over
to Degree.
• Press
3.Press
ENTER
WINDOW
.
. Use the window settings
shown.
Explain why the domain is set to
0  x  360.
4. Press
Y=
and enter sin x. Press
GRAPH
.
a) Describe the resulting graph.
b)Use the Maximum and Minimum operations of a graphing
calculator to find the maximum and minimum values of y.
At what values of x do these occur?
• Press 2nd [CALC] and select 4:maximum. Move the
cursor to locations for the left bound, right bound, and
guess, pressing ENTER after each.
• Press 2nd [CALC] and select 3:minimum. Move the
cursor to locations for the left bound, right bound,
and guess, pressing ENTER after each.
250 MHR Functions and Applications 11 • Chapter 5
c) What is the amplitude?
d)What would the graph look like if the domain were
extended to 720°?
5. Generate a table of values. Press 2nd
[TBLSET] and use the settings shown.
Then, press
2nd
[TABLE].
Scroll down and view the entire set of
ordered pairs for x-values between 0 and
360.
6.Describe each of the following for the sine function:
a) the amplitude
b) the period
c) the x-intercepts
d) the y-intercepts
e) the domain
f) the range
g)the intervals in which the function is increasing
and decreasing
Key Concepts
• A sine curve can be generated from a circle by plotting the rotational angle  and the
value of sin  as (x, y)  (, sin ).
y
y
P
sin �
(x, y) = (�, sin �)
�
0
x
0
x
• It is appropriate to use either x or  to represent the angle on the horizontal axis.
• For the wave generated by y  sin x, the period is 360° and the amplitude is 1.
• The domain of y  sin x is {x ∈ R}.
• The range of y  sin x is {y ∈ R | 1  y  1}.
5.3 Investigate the Sine Function • MHR 251
Communicate Your Understanding
C1 A clock pendulum swings back and forth with an amplitude
of 1 cm.
a) In order to generate a sine curve, what measurements should
be used for the horizontal and vertical axes?
b) Explain why the sine function would be an
amplitude
appropriate model.
C2 There is a simple method for sketching a graph of y  sin x. On a sheet of graph paper,
make an x-axis with the scale marked in increments of 30°. Mark the scale of the y-axis in
increments of 0.5. Plot the points (0, 0), (30, 0.5), (90, 1), (150, 0.5), (180, 0), (210, 0.5), and
so on, to (360, 0). Join the points with a smooth curve. Does this make sketching the graph
easier or harder?
Practise
2. Draw a sketch of y  sin x for one period.
A
1. Consider the unit circle discussed in
Section 5.2, Example 3. Let  represent the
angle, rotating counterclockwise from the
positive x-axis, and let y  sin .
a) What would be an appropriate scale for
the -axis?
b) Plot the ordered pairs (, y) for
0°    360°.
a) Locate all the points where y  1 and
give the values of x.
b) Locate all the points where y  0.5 and
give the values of x.
c) Locate all the points where y  1 and
give the values of x.
d) Locate all the points where y  0 and
give the values of x.
3. a) Copy this graph.
y
y
1
0
�
0
90°
180° 270° 360°
x
–1
c) How does the graph compare to your
graph in Investigate B?
d) What are the domain and range?
252 MHR Functions and Applications 11 • Chapter 5
b) Extend the graph for two more periods.
Label all intercepts and all maximum
and minimum points.
4. The minute hand of a clock is 10 cm
7. An emergency
in length and begins by pointing at the
number 9 on the face of the clock.
radio is designed
to function in cases
of power outages.
A common feature
of such a radio is a
hand-cranked
electrical generator
along with a
rechargeable battery to store the energy.
Define the length of the hand crank as
1 unit and the direction of rotation as
counterclockwise from a horizontal
position.
a) Draw this clock face and collect data
representing the distance of the tip
of the minute hand above or below
the level of 9 on the face of the clock,
relative to the angle of rotation.
b) Use your data to sketch a graph of the
angle versus the distance above or below
the number 9.
c) How does this graph compare to that of
y  sin ?
a) What is the height of the hand crank
Connect and Apply
relative to its centre of rotation after it
rotates 30°?
B
b) What is the height of the hand crank
5. Refer to Section 5.2, questions 9 and 10,
relative to its centre of rotation after it
rotates 135°?
in which you found coterminal angles as
values of  for given values of sin .
c) How many degrees has the hand crank
a) Plot your answers as ordered pairs (, y),
rotated through when its height is
0.2 units relative to its centre of
rotation?
where y  sin .
b) Join the points with a smooth curve.
c) How are the points above the -axis
related to the points below the -axis?
6. Chapter Problem Simulate a periodic
function by rotating your arm in a circular
motion with your shoulder as the centre of
rotation. Begin with your arm stretched out
in front of you.
a) Sketch a graph of the height of your
hand relative to the height of your
shoulder as a function of the rotational
angle from 0° to 720°. Define the length
of your arm as one unit of length.
b) What function would define this graph?
c) During which intervals is your hand
height increasing? Decreasing?
Extend
C
8. a) Find an equation of the line that passes
through all the maximum points of
y  sin x.
b) Find an equation of the line that passes
though the points on y  sin x where
x  0° and x  90°.
9. Consider the function y  cos .
a) Graph the function for two periods.
b) Compare the graphs of y  sin  and
y  cos  .
c) Determine all values of  where
sin   cos , for 0°    360°.
5.3 Investigate the Sine Function • MHR 253
5.4
Investigate Transformations of
Sine Curves
Not all sine curves can be modelled
using y  sin x. Often the amplitude
is greater than 1, as in the case of
the sound wave displayed on the
oscilloscope pictured here (the louder
the sound, the greater the amplitude
of the sound wave becomes).
Typically, a rotating object, such
as a Ferris wheel or a crank, has its
centre above the ground. Also, it
may not start its circular motion
from a horizontal position. Each of
these situations can be modelled by
transforming the sine curve.
Investigate A
C^^[b
•grid paper
•graphing calculator
How do transformations of the graph of y = sin x affect the
equation?
A: Vertical Translations
1.Clear all functions from the Y= editor and ensure all stats
plots are turned off. Then, make sure your graphing
calculator is in Degree mode.
Technology Tip
You can change the
appearance of a line. The
line style is displayed to the
left of the equation in the
Y= editor.
• Press Y = . Cursor left
to the slanted line beside
the equation.
• Press ENTER repeatedly
to choose one of the
seven options.
• Press GRAPH .
2.Set up the domain and the range.
Press WINDOW and use the settings shown.
3.Graph the function y  sin x as Y1.
• Press
Y=
• Press
GRAPH
. Beside Y1=, enter sin(x).
.
4.Enter y  sin(x)  2 as Y2 and y  sin(x)  3 as Y3.
Press
GRAPH
.
5. a)Sketch all three graphs on the same set of axes.
254 MHR Functions and Applications 11 • Chapter 5
b)Compare your graphs of
y  sin x  2 and y  sin x  3
to the graph of y  sin x. Consider
the period, amplitude, domain,
and range, as well as the horizontal
line that is halfway between the
maximum and minimum values
(the horizontal axis of the wave).
y
period
amplitude
horizontal
axis
x
6. Predict the graphs of y  sin x  4 and y  sin x  1. Sketch
the graphs on the same set of axes. Check your answers using a
graphing calculator.
7.Describe the effect of c in y  sin x  c on the graph of y  sin x.
B: Horizontal Translations or Phase Shifts
phase shift
•a horizontal translation of a
trigonometric function
1. a) Clear all the functions, except y  sin x.
b) Graph the functions y  sin(x  90°) and y  sin(x  90°).
2. a) Sketch all three graphs on the same set of axes.
b)Compare your graphs of y  sin(x  90°) and y  sin(x  90°)
to the graph of y  sin x. Consider the period, amplitude,
domain, range, and horizontal axis.
3.Predict the graphs of y  sin(x  30°) and y  sin(x  30°).
Sketch the graphs on the same set of axes. Check your
answers using a graphing calculator.
4. Describe the effect of d in y  sin(x  d) on the graph of y  sin x.
C: Vertical Stretches or Compressions
1. a) Clear all the functions except y  sin x.
1 ​ sin x.
b) Graph the functions y  2 sin x and y  ​ _
2
2. a) Sketch all three graphs on the same set of axes.
1 ​ sin x to the
b)Compare your graphs of y  2 sin x and y  ​ _
2
graph of y  sin x. Consider the period, amplitude, domain,
range, and horizontal axis.
1 ​ sin x. Sketch the
3.Predict the graphs of y  3 sin x and y  ​ _
3
graphs on the same set of axes. Check your answers using a
graphing calculator.
4.Describe the effect of a in y  a sin x, if a is positive, on the
graph of y  sin x.
5.4 Investigate Transformations of Sine Curves • MHR 255
D: Vertical Reflections and Stretches or Compressions
1. a) Clear all the functions, except y  sin x.
b) Graph the functions y  sin x and y  3 sin x.
2. a) Sketch all three graphs on the same set of axes.
b)Compare your graphs of y  sin x and y  3 sin x to the
graph of y  sin x. Consider the period, amplitude, domain,
range, and horizontal axis.
1 ​ sin x. Sketch the
3. Predict the graphs of y  2 sinx and y  ​ _
2
graphs on the same set of axes. Check your answers using a
graphing calculator.
5. Describe the effect of a in y  a sin x, if a is negative, on the
graph of y  sin x.
Investigate B
C^^[b
•CBR™ (calculator-based
rangefinder)
•chalk
•graphing calculator
•grid paper
•link cable
How can you relate a distance-time graph to the sine function?
1.Use chalk to mark a circle with radius 1 m on the floor, so that
the closest point is 2 m from a wall.
2.Connect a CBR™ to a graphing calculator.
3.Set the calculator to record data from the CBR™:
• Press
ENTER
APPS
, select CBL/CBR, and press
.
• Select 3:RANGER, press ENTER , and
select 1:SETUP/SAMPLE.
• Ensure your calculator settings match
those shown.
• Cursor up to START NOW and press
ENTER
.
4. Position yourself at the rightmost
point on your circle (3:00). Hold the
rangefinder with its sensor pointed at
the wall. Press ENTER and begin
walking at a constant speed in a
counterclockwise direction along the
circle for 15 s. Be sure to keep the
CBR™ pointed at the wall at all times.
5.Copy the graph onto grid paper, with time on the horizontal axis
and distance on the vertical axis.
256 MHR Functions and Applications 11 • Chapter 5
6.Explain why the distance from the wall to the CBR™
would provide a graph of a sine function.
7.How is your graph related to the graph of y  sin x?
8. Find an equation for the function relating the distance
from the wall to the angle of rotation.
9.Repeat the experiment walking at the same speed but
with the following changes.
a)Begin walking from the top of the circle (12:00). How
does the graph change? What would be the resulting
equation?
b)Walk a circle with radius 0.5 m centred inside the
original circle. How does the graph change? What
would be the resulting equation?
c)Draw a new circle with radius 1.0 m on the floor, so
that the closest point is 1.5 m from a wall. How does the
graph change? What would be the resulting equation?
Example 1
Sketch a Graph of a Transformation of y = sin x
Sketch a graph of each function for one cycle. Determine the
period, amplitude, phase shift, domain, range, and equation of
the horizontal axis of the cycle.
a) y  sin x  1
b) y  4 sin x
_1 4
c) y  ​ ​ sin x
d) y  sin(x  30°)
Solution
a) The graph of
y  sin x  1 is the
graph of y  sin x
translated upward
1 unit.
period  360°
amplitude  1
y
2
y = sin x + 1
y=1
1
0
90°
180°
–1
270°
360°
x
y = sin x
phase shift  0°
domain  {x ∈ R | 0°  x  360°}
range  {y ∈ R | 0  y  2}
The horizontal axis has equation y  1.
5.4 Investigate Transformations of Sine Curves • MHR 257
b) The graph of
y
4
y  4 sin x is the graph
of y  sin x stretched
vertically by a factor
of 4.
y = 4 sin x
2
0
period  360°
90°
180°
–2
amplitude  4
270°
y = sin x
360°
x
–4
phase shift  0°
domain  {x ∈ R | 0°  x  360°}
range  {y ∈ R | 4  y  4}
The horizontal axis has equation y  0.
c) The graph of
y
_1
y  ​ ​ sin x is the
1.0
4
graph of y  sin x
reflected in the x-axis and
compressed vertically by
1 ​ .
a factor of ​ _
4
period  360°
1 ​ amplitude  ​ _
4
phase shift  0°
1
y = – sin x
4
0.5
0
90°
180°
270°
360°
x
–0.5
y = sin x
–1.0
domain  {x ∈ R | 0°  x  360°}
{ _1 4
_1 4}
range  ​ y ∈ R | ​ ​  y  ​ ​ ​
The horizontal axis has equation y  0.
d) The graph of
y
2
y  sin(x  30°) is
the graph of y  sin x
translated to the right
by 30°.
period  360°
amplitude  1
1
0
y = sin (x – 30°)
90°
–1
phase shift  30° to the right
domain  {x ∈ R | 30°  x  390°}
range  {y ∈ R | 1  y  1}
The horizontal axis has equation y  0.
258 MHR Functions and Applications 11 • Chapter 5
180°
y = sin x
270°
360°
x
Example 2
Write an Equation From a Graph
Write an equation for each sine function.
a)
y
0
180°
360°
x
540°
y = –2
90°
180°
270°
–2
–4
b)
y
8
4
0
x
–4
–8
c)
y
1
–180°
0
–1
180°
x
(45°, 0)
Solution
a) Since the equation of the horizontal axis is y  2, this is the
graph of y  sin x translated downward 2 units.
The equation of the sine function is y  sin x  2.
b) Since the amplitude is greater than 1, the graph of y  sin x has
been vertically stretched by a factor of a. Find a, which is the
same as the amplitude.
maximum value
 ​
minimum value
amplitude  ​ _______
2
8  (8)
 ​ __
​ 2
16
 ​ _ ​ 2
8
The equation of the sine function is y  8 sin x.
c) Since the point (0°, 0) on the graph of y  sin x is now located at
(45°, 0), this is the graph of y  sin x translated to the right by 45°.
The equation of the sine function is y  sin(x  45°).
5.4 Investigate Transformations of Sine Curves • MHR 259
Key Concepts
Function
Transformation
Amplitude
and Period
Domain and Range
y = a sin x
a0
• vertical stretch of factor a if amplitude = a
period = 360°
domain = {x  R}
range = {y  R | –a  y  a}
y = a sin x
a0
• reflection in the x-axis
• vertical stretch of factor –a if amplitude = –a
period = 360°
domain = {x  R}
range = {y  R | a  y  –a}
• vertical translation of c units
• equation of horizontal axis is
amplitude = 1
period = 360°
domain = {x  R}
range = {y  R | c – 1  y  c + 1}
• horizontal translation of d units
amplitude = 1
period = 360°
domain = {x  R}
range = {y  R | –1  y ≤ 1}
a1
• vertical compression of factor a
if 0  a  1
a  –1
• vertical compression of factor –a if –1  a  0
y = sin x + c
y=c
y = sin(x – d)
•
(to the right if d  0; to the left
if d  0)
also called a phase shift
Communicate Your Understanding
C1 Describe the transformation from y  sin x that is needed to graph each of the following.
a) y  sin x
b) y  sin x  7
c) y  sin(x  40°)
C2 When graphing parabolas of the form y  a(x  h)2  k,
you considered transformations from the graph of y  x2.
Connecting
8
y = a(x – h)2 + k
4
2
–2
0
Reflecting
Selecting Tools and
Computational Strategies
6
–4
Communicating
Problem Solving
y
y = x2
Reasoning and Proving
Representing
2
4
6
x
Explain how this relates to how you use
transformations to graph sine functions.
260 MHR Functions and Applications 11 • Chapter 5
Practise
c) How would the graph change if the
A
circle was moved further from the wall
so that the closest point was 2.5 m from
a wall? What would be the resulting
equation?
For help with questions 1 to 3, refer to Example 1.
1. Sketch a graph of each function for
0°  x  360°. Determine the period,
amplitude, domain, and range.
d) How would the graph change if you
walked in a clockwise direction? What
would be the resulting equation?
a) y  6 sin x
_1 b) y  ​ ​ sin x
3
c) y  2 sin x
_1
d) y  ​ ​ sin x
2
2. Compare the graphs of each pair of
For help with questions 5 and 6, refer to
Example 2.
5. Write an equation for each sine function.
a)
functions for 0°  x  360°. Determine the
period, amplitude, domain, and range, as
well as the equation of the horizontal axis.
y
4
y=3
2
a) y  sin x  6 and y  sin x  6
b) y  sin x  1.5 and y  sin x  1.5
3. Compare the graphs of each pair of
functions for 0°  x  360°. Determine the
period, amplitude, phase shift, domain,
and range.
0
b)
360°
540°
90°
180°
270°
x
y
2
a) y  sin(x  60°) and y  sin(x  60°)
0
b) y  sin(x  270°) and y  sin(x  90°)
180°
x
–2
4. Refer to Investigate B, in which you walked
counterclockwise in a circle, beginning at a
point equivalent to 3:00 on a clock. Then,
you graphed the results.
c)
y
1 (90°, 0)
a) How would that graph change if you
instead began walking from the leftmost
point of the circle? What would be the
resulting equation?
–180°
0
180°
x
–1
b) How would the graph change if your
circle was 2 m in radius? What would
be the resulting equation?
5.4 Investigate Transformations of Sine Curves • MHR 261
6. Write an equation for each sine function.
a)
9. Graph one cycle of each function. Label
the x-intercepts, the maximum points,
the minimum points , and the equation of
the horizontal axis. Write the domain and
range of the cycle.
y
0
180°
360°
540°
x
–2
a) f(x)  0.4 sin x
y = –3.5
b) f(x)  4 sin x
–4
c) f(x)  sin x  5
b)
d) f(x)  sin x  8
y
e) f(x)  sin(x  45°)
2
f) f(x)  sin(x  90°)
0
90°
180°
270°
x
B
10. Without graphing, consider each sine
–2
c)
function. Identify the period, amplitude,
phase shift, domain, and range, as well as
the equation of the horizontal axis.
y
a) y  sin x  30 b) y  sin(x  30)
1
(–150°, 0)
0
–180°
11. For each of the functions, find the
180°
x
–1
Connect and Apply
7. Draw a sketch of y  2 sin x for one period.
a) Locate all the points where y  2 and
give the values of x.
b) Locate all the points where y  2 and
give the values of x.
8. Draw a sketch of y  sin(x  90°) for
one period.
a) Locate all the points where y  1 and
give the values of x.
b) Locate all the points where y  0 and
give the values of x.
coordinates of the maximum and minimum
points.
a) y  sin x
b) y  sin x  12
c) y  10 sin x
d) y  sin(x  60°)
12. a) Graph both f(x)  sin(x  180°) and
f(x)  sin(x  180°).
b) What do you notice about the two
graphs? Explain.
13. Write an equation for each sine function.
Indicate the intervals in which the function
is increasing and decreasing over one period.
a) amplitude  10, horizontal axis along
the x-axis
b) amplitude  1, horizontal axis along
y  8
c) amplitude  1, horizontal axis along the
x-axis, phase shift of 40° to the left
d) amplitude  1, horizontal axis along the
x-axis, phase shift of 100° to the right
262 MHR Functions and Applications 11 • Chapter 5
14. Chapter Problem A physiotherapist
establishes a treatment plan for a patient
that includes the use of an exercise bicycle.
The graph shows the height of a bicycle
pedal above its crank arm’s horizontal
position, relative to the rotational angle, as
the patient is pedalling it.
Achievement Check
16. A paddleboat’s wheel has radius 1 m. Its
axle is situated at the surface of the water.
a) Sketch a graph
to represent
the height
of a paddle
that begins at
the surface
of the water
and rotates
clockwise.
y
10
0
180°
360°
540°
x
–10
b) Determine an equation that represents
the height of the paddle relative to the
angle that the wheel spoke forms with
the horizontal.
c) How would the graph and equation
change if the wheel rotated
counterclockwise?
crank
arm
d) How would the graph and equation
change if the wheel was 2 m in radius?
pedal
e) Often, machinery uses periodic motion
to turn gears or generate electricity.
Think of a windmill or a mill wheel
that uses water to turn. Give some more
examples of machinery that could be
modelled with a periodic sine function.
In what ways would the equations be
similar?
a) Identify the period, amplitude, and
phase shift. What does each represent in
this situation?
b) Write an equation that represents the
function.
15. A clock is hanging on a wall with its centre
3.5 m above the floor. The minute hand is
20 cm in length and starts out pointing at 9
on the face of the clock.
a) Sketch a graph that represents the height
of the tip of the minute hand relative to
the angle it forms with the horizontal as
it rotates for one hour.
b) Determine an equation that represents
the height of the tip of the minute hand
with respect to the floor.
Extend
C
17. The graph of a sine curve passes through
the points (100°, 1), (190°, 0), (280°, 1).
Determine an equation that represents this
function.
18. A sine function of the form y  sin x  c
touches the x-axis but does not cross it.
Find all possible equations of this function.
19. A sine function of the form y  sin(x  d)
has the same graph as y  sin x. Find all
values of d.
5.4 Investigate Transformations of Sine Curves • MHR 263
5.5
Make Connections With Sine
Functions
How can you model periodic phenomena
such as the number of daylight hours in
a year or the volume of air in your lungs
as you breathe? In the previous section,
you investigated single transformations
of the sine function. However, accurate
models of periodic behaviours in the real
world typically involve combinations of
transformations of the sine function.
Investigate
C^^[b
How can you model the number of daylight hours?
•graphing calculator
The number of daylight hours, D, in Windsor, Ontario, on the
nth day of the year can be estimated using the equation
D(n)  2.9607sin(0.9863n  77.8374)  12.0318.
Technology Tip
You can find the number
of days between dates
in the format DD.MM.YY
using the dbd( function. For
example, find the number
of days between January 1,
2007, and May 13, 2007,
as follows:
• Press 2nd [CATALOG]
and select dbd(.
• Enter 0101.07. Press
. Enter 1305.07
ENTER .
and press )
May 13, is day 132 in 2007.
1. Use a graphing calculator to graph this function for two years.
Think about the appropriate window settings before you start.
Make sure your calculator is in Degree mode.
2.Predict when the maximum number of hours will occur.
Use the Maximum operation of the graphing calculator
to check. When does this occur?
3.Predict when the minimum number of hours will occur.
Check your prediction using the Minimum operation of
the graphing calculator. When does this occur?
4.When is the number of daylight hours increasing? Decreasing?
5.Identify the horizontal axis of the graph.
What does it represent?
6.What is the period of this graph? Explain.
264 MHR Functions and Applications 11 • Chapter 5
Model a Ride on a Ferris Wheel
Example
As a Ferris wheel turns, the height, h, in
metres, that a rider is above the ground is
given by the equation
h()  6sin(  90°)  8, where  is the
rotational angle after the ride begins.
Reasoning and Proving
Representing
Communicating
Problem Solving
Connecting
Reflecting
Selecting Tools and
Computational Strategies
a) Use a graphing calculator to graph this
function for two rotations.
b) Explain why a sine function is used to model the rider’s height.
c) What is the initial height of the rider?
d) What is the height of the rider after a rotation of 300°?
e) What is the maximum height of the rider?
Solution
a) Ensure that your graphing calculator is in
Degree mode. Press WINDOW and set Xmin  0
and Xmax  720. In the Y= editor, enter
6 sin(x  90°)  8. To make sure the range
is appropriate, press ZOOM and select 0:ZoomFit.
b) The position of a rider rotating around
the circular Ferris wheel repeats every 360°, so a sine function is
appropriate.
c) Using the TRACE feature, the y-intercept
is 2.
The initial height of the rider is 2 m.
2nd
[CALC] and select 1:value.
Enter 300 and press ENTER .
d) Press
The rider’s height after a rotation of 300° is
5 m.
[CALC] and select 4:maximum.
Move the cursor to locations for the left
bound, right bound, and guess, pressing
ENTER after each.
e) Press
2nd
The maximum height of the rider is 14 m.
5.5 Make Connections With Sine Functions • MHR 265
Key Concepts
• Accurate models of periodic behaviours in the real world often involve combinations of
transformations of the sine function.
• Technology, such as a graphing calculator, can be used to graph and analyse the function.
• Variables other than those for angles can be used in a sine function. An example would
be t for time.
Communicate Your Understanding
C1 Describe the steps involved in using a graphing calculator to determine the maximum or
minimum values of the function y  6 sin(2x  40)  10.
C2 Refer to the Investigate. Why would a sine function be used to model the number of
daylight hours?
Practise
Use a graphing calculator to answer each
question. Set the graphing calculator to Degree
mode.
A
For help with question 1, refer to the
Investigate.
1. The number of daylight hours, D, in
Edmonton, Alberta, on the nth day of the
year can be approximated by the equation
D(n)  4.6855sin(0.9856n  79.1491) 
12.1512.
For help with question 2, refer to the Example.
2. A wind turbine uses rotating blades to
produce electricity. The equation
h()  8.5 sin(  180°)  30 can be used
to find the height, h, in metres, of a point
on a given blade, where  is the angle the
blade makes with the horizontal.
a) Graph the function. Set Xmin  0 and
Xmax  720, and use ZoomFit to view
the graph. Sketch the graph.
b) Explain why a sine function was used to
model the height of a point on a blade.
a) Graph the function for two years.
c) What was the initial height of the point?
b) Determine the maximum number of
d) What is the maximum height of the
daylight hours, and when it occurs.
c) Determine the minimum number of
daylight hours, and when it occurs.
d) Compare the number of daylight hours
in Edmonton to that in Windsor, season
by season.
e) Edmonton is situated at approximately
53° north latitude. What change would
need to be made to the equation for
daylight hours in Punta Arenas, Chile,
at 53° south latitude?
266 MHR Functions and Applications 11 • Chapter 5
point?
e) How high above the ground is the axis
of a blade?
Connect and Apply
7. Chapter Problem A respiratory therapist
evaluates a patient’s lung capacity. The
volume of air, V, in millilitres, in a patient’s
lungs during normal breathing is given by
V(t)  500 sin(60t)  2400, where t is the
time, in seconds.
3. A satellite orbits Earth such that its
displacement (distance north or south)
from the equator (ignoring altitude) is given
as y  7200 sin(1.43t  14.32), where t is
the time, in minutes, and y is the distance,
in kilometres.
a) Sketch a graph of this function for 15 s.
b) Determine the period and amplitude
a) Sketch a graph of the function for
from the graph.
500 min.
c) How would the graph change if the
b) What is the displacement from the
patient breathed faster?
equator after 1 h?
d) How would the graph change if the
patient took deeper breaths?
4. The electric current, i, in microamperes
(μA), in a circuit is given by the equation
i  4 sin(360t  11.5), where t is the time,
in seconds.
a) Sketch a graph of the function for 2 s.
b) What is the amplitude of the current?
8. The current, i , in amperes (A), passing
through a wire is given by the equation
_2
i  6 sin​ ​ ​ t  30 ​, where t is the time, in
3
seconds.
( )
a) Graph the function.
B
b) Determine the period, amplitude, and
5. A crank rotates such that the height of the
handle, h, in metres is given by
h()  2 sin(  90°), where  is the
rotational angle relative to the horizontal.
a) Sketch a graph of this function for two
rotations.
b) At what height was the handle when the
rotation began?
c) What was the position of the handle
relative to the horizontal, when the
rotation began?
6. The CN Tower in Toronto, Ontario is
approximately 554 m tall. In strong winds,
the top of the communication tower will
sway up to 2 m. On a particular day, the
displacement of the top of the tower in
metres, relative to the normal position, is
modelled by f(t)  0.9 sin(2t), where t is the
time, in minutes.
a) Sketch a graph that shows the swaying
of the top of the communication tower
over 8 h.
phase shift of the current from the
graph.
Extend
C
9. The displacement from the equator, D,
in kilometres, of an orbiting satellite can be
determined using the formula
D(t)  A sin(wt  p). Consider the graph
of D(t)  600 sin(200t  30), where t is the
time, in hours.
a) What significance do 600, 200, and 30
have in the equation?
b) Generalize by describing the significance
of A, w, and p.
10. Without the use of technology, sketch a
graph of each function for one period.
a) f (x)  2 sin(x)  3
b) f (x)  sin(x  180°)
_1 c) f (x)  ​ ​ sin(x  90°)  3
2
b) How wide is the sway?
c) What is the period of the sway?
5.5 Make Connections With Sine Functions • MHR 267
Chapter 5 Review
5.1 Periodic Functions, pages 232–238
1. Determine whether or not each graph is
periodic. If it is, determine the period,
amplitude, domain, and range.
a)
y
40
20
–2
0
2
4
6 x
–20
b)
3. An angle is in standard position with its
terminal point, P, given. Find the radius of
the circle in exact form and the measure of
the angle to the nearest tenth of a degree.
a) P(4, 3)
b) P(12, 5)
c) P(7, 8)
d) P(2, 1)
4. A terminal point, P(x, y), on the unit circle
forms the given angle in standard position.
Find the coordinates of point P, to three
decimal places.
y
a) 60°
b) 135°
c) 180°
4
d) 166° e) 290°
f) 320°
5.3 Investigate the Sine Function,
pages 248–253
2
0
5
10
15 x
–2
2. Ocean tides rise and fall at sites around
the world. The table shows the depth of
water at one location for a 2-day period.
Time
Depth (m)
3:42
3.16 low tide
9:48
12.13 high tide
16:14
2.85 low tide
22:17
12.01 high tide
4:38
5.2 Circles and the Sine Ratio,
pages 239–247
2.90 low tide
10:40
12.32 high tide
17:04
2.61 low tide
23:05
12.32 high tide
a) Sketch a time graph of the tide depths.
b) Estimate the period and the amplitude
of the tide cycle.
c) Estimate the mean depth of the water.
5. A crank handle is used to extend and
retract an awning. The radius of the
crank handle is 1 unit and the direction it
rotates is counterclockwise starting from a
horizontal position. Round answers to the
nearest tenth of a unit when necessary.
a) What is the height of the handle after it
rotates 60°?
b) What is the height of the handle after it
rotates 180°?
c) What is the height of the handle after it
rotates 2000°?
d) Find two rotational angles at which the
handle has height of 0.5 units.
5.4 Investigate Transformations of Sine
Curves, pages 254–263
6. Sketch a graph of each function for one
cycle. Determine the period, amplitude,
phase shift, domain, range, and the
equation of the horizontal axis.
_1
a) y  5 sin x
b) y  ​ ​ sin x
4
c) y  sin x  12 d) y  sin x  2.5
e) y  sin(x  60°)f) y  sin(x  90°)
268 MHR Functions and Applications • Chapter 5
7. Write an equation for each sine function.
a)
y
1
0
8. Use Technology According to the
180°
360°
–1
x
540°
y = –1
–2
b)
y
2
0
90°
180°
270°
x
–2
c)
5.5 Make Connections With Sine Functions,
pages 264–267
0
( ( )
)
( )
9. Use Technology The height of a rider on a
Ferris wheel, in metres, can be modelled
using the function h()  10 sin(  90°)  12,
where  is the angle of rotation.
y
1
biorhythm theory, three cycles affect
people’s lives, giving them favourable and
non-favourable days. The physical cycle
360 ​ can be modelled as y  sin​ ​ _
t ​, where t
23
represents a person’s age, in days.
Similarly, the emotional cycle can be
90 ​ t ​and the
modelled as y  sin​ ​ _
7
120 ​ intellectual cycle as y  sin​ ​ _
t ​. Use a
11
graphing calculator to compare the three
cycles for 0  t  60.
a) Use a graphing calculator to graph the
(45°, 0)
90°
function.
180°
270°
x
–1
b) What is the radius of the Ferris wheel?
c) At what height was the rider when the
ride began?
d) How would the function and the graph
change if the Ferris wheel turned in the
opposite direction?
Chapter Problem Wrap-Up
Ultrasound is a medical imaging technique that has both diagnostic and therapeutic applications.
A medical imaging technician uses an ultrasound machine on a patient. The signal produced can
be defined by the equation I(t)  5 sin(1 100 000t  23), where I represents the sound radiation, in
nanoWatts (nW), and t represents the time, in seconds.
a) With the help of a graphing calculator, sketch a graph of this relation for 0  t  0.0007.
b) What is the maximum and minimum radiation produced by this ultrasound machine?
c) What is the period of the sound radiation? How many cycles occur in 1 s?
d) How does this graph compare to a wave with equation I(t)  sin t?
e) The technician adjusts the machine’s settings so that the signal is now defined by
I(t)  8 sin(1 500 000t  23). Describe the adjustments that were made.
Chapter 5 Review • MHR 269
Chapter 5 Practice Test
For questions 1 to 4, select the best answer.
1. What is the period of the graph shown?
y
2
6. Determine the measure of angle  in standard
position, to one decimal place.
a) sin   0.5, if  is in the second quadrant
b) cos   0.4558, if  is in the third
quadrant
0
180°
360°
540°
x
c) tan   2.2361, if  is in the fourth
quadrant
–2
d) sin   0.0960, if  is in the second
quadrant
A 180°
7. Draw a sketch of y  sin x for one period.
B 2°
a) Locate all the points where y  1 and
C 90°
give the values of x.
D 360°
2. The value of d in y  sin(x  d) refers to
A the phase shift
b) Locate all the points where y  0.7071
and give the values of x.
c) Locate all the points where y  0.5
B the amplitude
and give the values of x.
C the vertical translation
8. Sketch a graph of each sine function,
D the period
showing two cycles.
3. An angle  is in standard position with
terminal arm in the third quadrant. Which
statement is true?
A Sin  is positive and cos  is negative.
B Sin  is negative and cos  is negative.
C Sin  is positive and cos  is positive.
D Tan  is negative.
a) f(x)  3 sin x
b) f(x)  sin(x  60°)
c) f(x)  sin x  5
9. Determine an equation for each sine
function.
x-axis
4. Which angle is not coterminal with 150°?
_1 a) amplitude  ​ ​ , reflected in the
b)
A 510°
B 870°
2
y
2
y=1
1
C 1230°
0
D 330°
5. Sketch each angle in standard position.
Then, find two coterminal angles for each
given angle.
a) 30°
b) 65°
c) 160°
d) 180°
e) 200°
f) 305°
g) 193°
h) 270°
c)
360°
540°
x
180°
360°
540°
x
y
8
4
0
–4
–8
270 MHR Functions and Applications 11 • Chapter 5
180°
10. Write an equation for each sine function.
Indicate the intervals in which the function
is increasing and decreasing over one
period.
a) amplitude  5, horizontal axis along the
x-axis
13. Use Technology The height, h, in metres,
of a point on a boat’s propeller, relative to the
surface of the water is defined by
h()  0.12 sin(  90°)  0.43, where  is
the rotational angle, in degrees.
a) With the help of a graphing calculator,
b) amplitude  1, horizontal axis along
y3
c) amplitude  1, horizontal axis along the
x-axis, phase shift of 90° to the right
d) amplitude  1, horizontal axis along the
x-axis, phase shift of 30° to the left
11. A toy tractor wheel has radius of 1 cm
and rotates counterclockwise. A stone is
embedded in the wheel at the rightmost
point, as shown. Describe the change to the
sine graph in each situation.
sketch a graph of the function for three
rotations of the propeller.
b) What would be the maximum and
minimum depths of the propeller?
c) Determine the depth of the given point
on the propeller after rotations of 240°
and 600°. Explain the results.
Achievement Check
14. Use Technology The number of people, in
millions, who used public transit in a large
city during any given month can be modelled
by the function f(x)  2.3 sin(30x  30)  4.7,
where x represents the month, with
January  1, February  2, and so on.
a) Use a graphing calculator to graph the
stone
function for one year.
b) Approximately how many people used
public transit in August?
a) The wheel is 2 cm in radius.
b) The stone begins in the uppermost
position.
c) The wheel rotates clockwise.
12. Use Technology The distance, d, that the
moon is from Earth, in kilometres, can be
estimated using the function
d(t)  25 500 sin(13.211t  90)  381 500,
where t is the time in days after the perigee
(the day when the moon is closest to Earth).
a) Use a graphing calculator graph the
c) During which month was ridership at its
highest level?
d) In which two
months did
about
4 000 000
people use
public transit?
Reasoning and Proving
Representing
Communicating
Problem Solving
Connecting
Reflecting
Selecting Tools and
Computational Strategies
e) In what ways
might global warming
affect this model?
function for two cycles.
b) Find the maximum and minimum
distances the moon is from Earth.
c) How many days is one cycle of the
moon?
Chapter 5 Practice Test • MHR 271
Chapter 5 Task
Model the Rotation of the Earth on its Polar Axis
Earth facts:
• The average distance from the sun to the centre of the Earth is
150 000 000 km.
• The average time for Earth to complete 1 full rotation is 23 h,
56 min and 4 s. This is known as a sidereal day. Since the Earth is also
revolving around the sun, a day is 24 h long.
If the sun is level with the horizon line at 6 a.m. and rises above the
horizon line as the Earth rotates in the opposite direction, its position
relative to the horizon line can be modelled by a sine function with a
phase shift and a vertical stretch.
12 P.M.
10 A.M.
8 A.M.
150 000 000 km
60°
Position of the sun relative
to the horizon line
30°
Horizon line
150 000 000 km
Earth’s
rotation
6 A.M.
a) Create a table to record the sun’s distance relative to the horizon line.
Time of Day

sin 
Position Relative to Horizon
06:00
0º
0.0
0
08:00
30º
0.5
75 000 000
b) Plot a standard sine curve to represent the apparent movement of the
sun as the Earth rotates in the opposite direction.
c) Transfer the sine curve to a graph where the time of day is
represented by the x-axis and the sun’s position relative to the
horizon is represented by the y-axis.
272 MHR Functions and Applications 11 • Chapter 5
d) What does the minimum value of the sine curve from part c)
represent in the context of this situation? What does the maximum
value represent?
e) Determine the intervals for which the since curve from part c) is
increasing or decreasing.
f) Write an equation that represents your graph from part c).
Model of Earth and Sun
Create a three-dimensional model that shows how the apparent position of
the sun in the sky moves as the Earth rotates on its axis, modelling a sine
function.
Chapter 5 Task • MHR 273
Chapters 4 and 5 Review
Chapter 4 Trigonometry
4. When a cat is 7 m from a fence, the angle
1. Find the length of the indicated side, to the
nearest tenth of a unit.
b)
a)
C
E
5. Two scuba divers are 100 m apart at the
55°
8.5 cm
b
c
26°
A
B
c)
C
L
G
20°
1.8 m
K
J
E
2. Find the measure of both acute angles in
each triangle, to the nearest degree.
b)
E
T
9m
F
D
apart. Marta is 12.5 m from one post and
13.2 m from the other post. Within what
angle, to the nearest degree, must she kick
the soccer ball to score a goal?
Chapter 5: Sine Functions
7. Determine whether or not each graph is
11 cm
7 cm
same depth beneath the water’s surface.
Their boat is located between the two
divers at an angle of elevation of 60° from
diver A and 54° from diver B. How far is
each diver from their boat, to the nearest
metre?
6. The posts of a soccer goal are 7.3 m
k
f
64°
D
9.2 m
d)
2.1 cm
F
a)
of elevation to the top of the fence is 16°.
What is the angle of elevation when the cat
is 3 m from the fence? Round to the nearest
degree.
R
S
13 m
periodic. If it is, determine the period,
amplitude, domain, and range.
a)
y
3. Solve each triangle. Round your answers to
4
the nearest tenth of a unit, if necessary.
a)
b)
A
–4
N
0
4
8 x
4
8 x
–4
9.1 cm
C
43°
12.7 cm
L
67°
51°
2.3 m
M
b)
y
B
c)
4
d)
G
23 cm
Y
Z
–4
59 cm
4.0 m
62 cm
H
58°
3.2 m
F
X
274 MHR Functions and Applications 11 • Chapters 4 and 5 Review
0
–4
8. The table shows the number of
international travellers visiting Canada
from Europe quarterly from 2005 to 2006.
Quarterly
Period
Number of Travellers
(1000s)
March 2005
327
June 2005
607
September 2005
1034
December 2005
405
March 2006
319
June 2006
616
September 2006
996
December 2006
403
Source: Statistics Canada, CANSIM table 387-0004
a) Make a scatter plot of the data and draw
a curve of best fit.
b) Does this represent a periodic function?
Explain your reasoning.
c) What are the maximum and minimum
numbers of travellers? When do they
occur?
d) Predict the maximum number of
travellers for the year 2010. How
accurate is your prediction? Explain
your reasoning.
9. Which pairs of angles are coterminal?
Justify your answer.
a) 40° and 220°
b) 65° and 785°
c) 115° and 245° d) –35° and 235°
10. An angle is in standard position with
terminal point, P. Find the radius of the
circle, in exact form, and the measure of
the angle, to the nearest tenth of a degree.
Include a diagram illustrating each angle.
a) P(3, 4)
b) P(5, 2)
c) P(6, 8)
d) P(2, 1)
11. Determine the measure of angle  in
standard position, correct to one decimal
place.
a) sin   0.7660, if  is in the second
quadrant
b) sin   0.8910, if  is in the third
quadrant
c) tan   0.3640, if  is in the fourth
quadrant
d) cos   0.8660, if  is in the third
quadrant
12. Draw a sketch of y  sin x for one period.
a) Locate all the points where y  0 and
give the values of x.
b) Locate all the points where y  0.8660
and give the values of x.
c) Locate all the points where y  0.7071
and give the values of x.
13. Sketch a graph of each function for one
cycle. Determine the period, amplitude,
phase shift, domain, range, and the
equation of the horizontal axis.
a) y  sin x  3
b) y  2 sin x
1
_
c) y  ​ ​ sin x
d) y  sin(x  30°)
2
14. Use Technology The hawk population in a
particular region varies with the number of
rodents that inhabit that region. The hawk
population, h(t), is given by
h(t)  25 sin (45t)  250, where t is the
number of years that have passed since
1980.
a) Use a graphing calculator to graph the
function.
b) In the first cycle of this function, what
was the maximum number of hawks and
in which year did it occur?
c) In the first cycle of this function, what
was the minimum number of hawks and
in which year did it occur?
d) Determine the period of the graph.
Chapters 4 and 5 Review • MHR 275