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Mathematics C30
Module 1
Lesson 2
Mathematics C30
Radian Measure and Trigonometric Functions
67
Lesson 2
Mathematics C30
68
Lesson 2
Radian Measure and Trigonometric
Functions
Introduction
In Lesson 1 the unit for measurement of angles was the degree. It is common knowledge
that there are 360 degrees in a circle, 90 degrees in a right angle, and that each angle of
an equilateral triangle contains 60 degrees. Also, each degree is subdivided into 60
minutes, and each minute is subdivided into 60 seconds.
The Babylonians found it natural to divide circles into 360 degrees. They had a base-60
number system, and 6 times 60° or 360° was a convenient way to subdivide the circle. (We
use a base-10 system.) They knew that a hexagon inscribed in a circle had a perimeter
that is 6 times the radius of the circle. Therefore, if each central angle of the hexagon is
given the measure of 60°, the circle would have 6 × 60° which is equal to 360°.
r
r
60°
r
Another measure of angles is called the radian. It is found to be more convenient because
many formulas are easier to write and understand using radians.
In this lesson radian measure will be introduced and used in various applications.
Your scientific calculator is able to work in the radian mode, as well as in the
degree mode. You may want to investigate how to switch from degree mode
to radian mode, and vice versa, before you begin this lesson.
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Lesson 2
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Lesson 2
Objectives
After completing this lesson you will be able to

determine the radian measures of angles and convert from
radians to degrees, and vice versa

work with the trigonometric ratios using radian measure.

determine angular velocity and apply this concept to
problems involving rotation.

determine arc length and apply this to word problems.

define and illustrate the terms periodic function, amplitude,
domain, range, minimum value, maximum value,
translation, and sinusoidal functions.

state the range, period, amplitude, phase shift, minimum and
maximum values, and sketch the graphs of
a)
b)
c)
Mathematics C30
y = a sin(bx + c)
y = a cos(bx + c)
y = a tan(x + c)
71
y = d + a sin(bx + c)
y = d + a cos(bx + c)
y = d + a tan(x + c)
Lesson 2
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Lesson 2
2.1 Radian Measure of an Angle
The Unit Circle

y
Let the circle illustrated in this diagram have a
radius (r) equal to 1.

The circumference of this circle is 2r  21  2 .

A circle with radius 1 is known as a unit circle.
P
r= 1
O
x
In the unit circle the point P travels along a circular arc TP in a counterclockwise
direction.
y
P
r= 1
O
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When P has completed one journey around the circumference
of the unit circle, it has covered a distance equal to the length
x of the circumference or 2 .
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Lesson 2
Study the following diagrams. Notice that positive measures have been assigned to arcs
generated by a counterclockwise motion of P.
y
y
P
90°
180°
r= 1
P
T
x
T
The length of arc TP is  ,
since

2
1
2     .
2
x
The length of arc TP is
since
1
2    .
4
2

,
2
Radian Measure or Circular Measure
If the point P travels through a length of arc the same length as the radius, the angle
subtended by the arc is given the measure of 1 radian.
A
If the arc BA = the radius AO, then angle AOB has a measure
of 1 radian (1 rad).
O
1 rad
r= 1 B
x
Thus, on the unit circle, the length of the arc subtended by the
angle is the measure of the angle in radians.
Referring back to the two circles, a 180° angle has a radian measure of  radians

and a 90° angle has a radian measure of
radians.
2
On the unit circle, one radian is the measure of the angle at the centre of
a circle subtended by an arc that is equal in length to the radius of the
circle.
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Lesson 2
Relationship between Degrees and Radians
In degree measure, one revolution is 360°. In radian measure, for the unit circle, one
revolution is 2 radians, which is the circumference length.
Therefore, 2 rad is equivalent to 360°.
If both terms are divided by 2 , we get
2
rad is equivalent
2
1 rad is equivalent

360 
, or
2
180 
to
.
to

 180 
 180 
From this, it follows that 2 radians = 2 
 degrees, 3 radians = 3 
 degrees, etc.
  
  
To convert radians to degrees, multiply radians by
180 
.

When working with radian measure it is convenient to not write the word radian
after the measure. If no degree symbol is written after a number, it is understood
to be radians.
To convert degrees to radians, divide both terms by 360 to get
2
is equivalent
360
to, or
360 
360

radians is equivalent to 1°.
180

  
From this it follows that 2 degrees = 2 
 radians, 3 degrees =
 180 
To convert degrees to radians, multiply degrees by
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  
3
 radians, etc.
 180 

.
180 
Lesson 2
Example 1
i)
Convert the following radian measures to degrees.
a.
ii)
places).

6
b.
7
Convert the following degree measures to radians. (Round to 4 decimal
a.
45°
b.
75°
Solution:
i)
ii)
a.

6
radians
is equivalent
b.
7 radians
a.
45  is equivalent
b.
75  is equivalent
is equivalent

180 
 30 .
6

6
180  1260 
to 7 

 401 .07 .
to
to 45  

180 




 0 .7854 radians.
4

235 .5
to 75  

 1 .3090 radians.
180 
180
180 


It is possible to use your scientific calculator to convert from
radians to degrees and vice versa. The answers on the calculator
may be slightly different because of the value for  .
Throughout this lesson, questions involving  are calculated
using the  key on your calculator (rather than using 3.14).
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Lesson 2
Exercise 2.1
1.
In the figure, the angles are measured in the counterclockwise direction from the
positive x-axis. Complete the ordered pairs by converting to radian measure in
lowest terms, using  where possible for each of the common angles given in
degrees.
y
{90°,
{105°,
}
{120°,
{75°,
}
{60°,
}
{15°, /12}
}
{0°, 0} &
{360°, 2 }
{180°, }
{195°,
}
{225°,
x
{345°, 23 /12}
}
{330°,
}
{210°,
}
{240°,
{315°,
}
{255°,
}
{270°,
}
{300°,
}
{285°,
2.
In radians, the complement of  

is ***.
3
3.
In radians, the supplement of  
3
 is ***.
4
Mathematics C30
}
{30°,
}
{150°,
}
{45°,
}
{135°,
{165°,
}
}
77
}
}
Lesson 2
2.2 The Trigonometric Functions
The use of radians allows the trigonometric functions to be used for situations where angle
measure is not involved. For example, if t represents time in seconds, the expression
15 cos t could represent the height of a water wave at time t. In this case no angle
measure is involved.
Since the unit circle was used to define radians, whenever radian measure is used instead
of degrees, the sine, cosine, tangent, secant, cosecant, and cotangent functions are
sometimes called circular functions.
An expression like sin 5 means the sine of any angle whose radian measure is 5. The
equivalent degree measure can also be found.
 180  
5
  286 .4789 
  
Therefore, sin 5  sin 286 .4789    0 .9589 .
Example 1
Evaluate cos 30 and cos 30  .
Solution:
Determine cos 30. Since no degree symbol is given, the number is understood to be in
radians.
 180  
cos 30  cos 30 
  cos 1718 .9   0 .1543
  
Determine cos 30°.
cos 30  
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 0 .8660
2
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Lesson 2
With the calculator in radian mode:
cos 30  0 .1543
With the calculator in degree mode:
cos 30   0 .8660
Example 2
If 350 sin 2 t  represents the height in cm of a water wave at any time t in seconds,
find the height at 10 seconds.
Solution:
Determine the height of a water wave at 10 seconds.
h  350 sin 2  10 
h  350 sin 20
 180  
h  350 sin 20 

  
h  350 sin 1145 .9 
h  319 .49 cm
With the calculator in radian mode calculate:
h  350 sin 20
h  350 0 .9129 
h  319 .5 cm
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Lesson 2
Example 3
If tan  
3
, find  such that 0    2 .
3
Solution:

3

,  is the common angle 30° or .
6
3
Tangent is positive in the first quadrant.

The first solution is

Tangent is also positive in the third quadrant and any other solution would have the

same reference angle, .
6

The second solution is  

It is useful to recognize that if tan  
Mathematics C30

.
6
 7
 .
6 6
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Lesson 2
Example 4
Determine the x and y coordinates of the point where the unit circle intersects the
7
terminal ray of an angle in standard position with measure  .
6
Solution:

Remember, the radius of a unit circle is 1.
x
r
 Cosine is negative in Quadrant 3.
 cos  
Calculate x.
cos


7
x x
  x
6
r 1
7
x  cos 
6

x   cos
6
3
x
2
The reference angle is

.
6
The reference angle is

.
6
y
r
Sine is negative in Quadrant 3.
sin  
Calculate y.
sin
Therefore,
7
y y
   y
6
r 1
7
y  sin 
6

y   sin
6
1
y
2

3
1
P  x , y   P  
,   .
2
 2
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Lesson 2
All the topics that were studied in Lesson 1 can also be calculated with radian
measure. It is useful to be able to quickly recognize the radian measure of the
common angles and to give the exact values of the trigonometric functions of
these angles.

6

4

3
30°
45°
60°
The following exercises have questions similar to those in Lesson 1 but involve the use of
radians. In these exercises, you will become familiar with the use of radian measure of
common angles.
Exercise 2.2
1.
2.
Use your calculator to evaluate each to four decimal places and find x, where

0x .
2
a.
tan 14
f.
sin x  0 .4078
b.
tan 14
g.
cos x  0 .8196
c.
cos 1000 
h.
d.
sec 1000
i.
e.
cos 3
j.
tan x  3
1
tan x 
4
cot x  4
All the angles in this question are common or quadrantal. Give exact values or, if
undefined, state so.
a.
sin




, tan , cos , cot
6
4
3
2
b.
sec



, cos 0 , csc , tan
3
3
2
c.
cos , sin , csc , tan 
d.
csc
Mathematics C30
3
3
3
3
, tan
, sin
, cos
2
2
2
2
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Lesson 2
3.
For angles that are in the second quadrant, express in terms of the reference angle
with the appropriate + or – sign, and then give the exact value.
a.
b.
c.
d.
4.
2

3
5
csc 
6
3
cot 
4

cos
2
e.
cos
f.
g.
h.
For angles that are in the third quadrant, express in terms of the reference angle
with the appropriate + or – sign, and then give the exact value.
a.
b.
c.
d.
5.
2

3
5
tan 
6
3
cos 
4
5
sec 
6
sin
4

3
5
sin 
4
7
cos 
6
5
tan 
4
sin
4

3
7
csc 
6
7
tan 
6
3
cos 
2
e.
sec
f.
g.
h.
For angles that are in the fourth quadrant, express in terms of the reference angle
with the appropriate + or – sign, and then give the exact value.
a.
b.
c.
d.
5

3
7
sin 
4
11
cos

6
5
cos 
3
tan
Mathematics C30
5

3
7
cos 
4
11
cot

6
7
csc 
4
e.
sin
f.
g.
h.
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Lesson 2
6.
For each question, find all real numbers  , 0    2 .
a.
sin   1
b.
cos   1
c.
sin   0
d.
tan   0
e.
tan  is undefined
f.
csc  is undefined
g.
sin  
h.
cos   
i.
2
2
3
sin   
2
j.
1
2
1
2
cos   
2
, find  and cos  .
2
7.
If tan   1 and sin   
8.
Evaluate by giving the simplified exact values.
a.
b.
9.

3
7
5
sin   cos  cos 
4
4
4
4
7
2
7
2
csc  sec   sin  cos 
6
3
6
3
sin
Determine the coordinates of the point where the unit circle intersects the terminal
ray of an angle in standard position with the given measure.
a.
b.
c.
d.
3

2
17 
2
17

6
8

3

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Lesson 2
2.3 Arc Length and Angular Speed
Arc Length
In the unit circle (a circle with radius, r = 1),
radian measure of an angle is defined to be the
length of the arc of the circle subtended by the
angle.
r= 1

arc subt ended by 
Radian measure can also be determined by using
any circle other than the unit circle.
The following examples suggest that for any circle of radius r,
length of arc subtended
r
by 
 radian measure of 
For any circle of radius r, the circumference formula is:
C  2 r
C
 2
r
Divide by r.
For a 360° angle, the radian measure 2 is the ratio

C
.
r
Consider half the circumference.
1
1
C  2   r
2
2
1
C
2 
r

Consider one quarter the circumference,
1
1
C  2   r
4
4
1
C
4 
r
2
In each case
arc length
radius
Mathematics C30
 radians .
85
Lesson 2
Arc Length
For any circle of radius r
arc length
radius
or
 radians,
s

r
s
 ,
r
s  r
Example 1
If the radius of the earth is taken to be 6400 km, how many kilometres are there
between the 48 and 49 parallels of latitude?
Solution:
The difference is 49   48   1  .
Convert the degrees to radians.
  1
 1

Substitute into the formula.

180

rad.
180
s r
 

s  6400 km  1 

 180 
6400 
s
180
s  111 .7 km
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Lesson 2
Example 2
A point is 36 cm from the center of rotation and moves 70 cm in a circular path.
Through what angle has the point moved?
Solution:
Apply the formula s  r .
s  r
70 cm  36 cm  
70

36
  1 .9
The point moved through an angle of 1.9 radians.

In degrees,   109  .
Example 3
The wheels of a car have a radius of 40 cm. If the car travels 1 km, how many
rotations does one wheel make?
Solution:
Be sure the units are the same.


In travelling 1 km, the arc length is 1 km.
1 km = 100 000 cm
Apply the formula s  r  .
s r
100 000 cm  40 cm  
  2500
Determine the number of revolutions.
Mathematics C30
1 rev
x

2
2500
2500 1 
x
2
x  398 revolution s
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Lesson 2
Angular Speed
Linear Speed is defined to be the distance traveled per unit of time. In all cases in this
course, speed can be determined by dividing distance traveled by the time it takes to
travel that distance.
Similarly, angular speed is defined to be the amount of rotation (either radians or
revolutions) per unit of time.
In this section the objects will be traveling in a circular path and distance will be the
length of the arc over which an object has travelled.
Example 4
In 3 seconds a wheel of radius 20 cm makes 10 revolutions 10  2 radians.
a.
b.
c.
How many centimetres does a point on the edge of the wheel travel?
What is the velocity of a point at the edge of the wheel?
Find the angular speed of the wheel.
Solution:
a.
Use the formula s  r 
S  r
b.
Determine the linear speed.
v
s  20 cm 10  2  
s  1256 .6 cm
.
c.
1256 .6 cm
3s
v  418 .9 cm/s
v
10 rev
 3 .3 rev/s  in revolutions
3s
10  2 
 20 .9 / s
 in radians (as usual
3s
the term “radians” is not
included.)
Determine the angular speed.
Mathematics C30
s
t
88
Lesson 2
Speed Formula
Given:
arc length = (radius)(radians)
s r
Divide both sides by time (t).
s

 r , where
t
t
•
•
•
Therefore:
s
= linear speed (v)
t
r = radius (r)

= angular speed (a)
t
vra
Example 5
a.
b.
Convert 25 revolutions to radians.
Convert 16 radians to revolutions.
Solution:
a.
25 revolutions is 2 radians/re v 25 rev   157 radians
b.
16 radians is
16  radians
 8 revolution s .
2  radians / rev.
Radians and Revolutions
radians
 rev  radians
rev
radians
radians
rev
 radians 
rev
radians
 rev
Notice that the units follow algebra rules of cancellation.
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Lesson 2
Exercise 2.3
1.
Determine the value of the indicated variable. In each case use the proper units.
•
•
•
a.
b.
c.
d.
s = arc length
r = radius
 = radians
5
4

s = ?, r = 10 cm,  
3
s = 27 cm, r = ?,  
s = 35 cm, r = 17.5 cm,   ?
s = 2.1 m, r = ?,   75 
2.
A pendulum 100 cm long swings through an angle of 15°. Find the length of arc
that the tip of the pendulum makes to the nearest centimetre.
3.
The earth has a 6400 km radius and makes one revolution every 24 hours. What is
the linear speed of a point on the equator in km/h?
4.
A bicycle wheel has a 70 cm diameter and a cyclist travels 20 km in one hour on this
bicycle. Find the angular speed of the wheel in revolutions/s.
5.
A Ferris wheel at an amusement park has a radius of 8 m. If it completes two
revolutions in 11 seconds, determine the angular velocity of the Ferris wheel in
radians per second.
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90
Lesson 2
6.
The angular speed of a car wheel is 70 radians per second. If the radius of the tire
is 28 cm, how far will the car travel in 30 seconds? Express the answer to the
nearest metre.
7.
A wheel revolves at 120 radians per minute.
a.
b.
c.
What is the angular velocity in radians per second?
What is the angular velocity in radians per hour?
What is the angular velocity in revolutions per minute?
2.4 Graphs of Trigonometric Functions
Intervals
The domain and range of a function are usually intervals on the real line. Before
beginning the study of the graphs of trigonometric functions, a list of various kinds of
intervals is given.
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Lesson 2
The solid dot in the graph means that the end point is included in the interval, and the
empty circle means that it is not included.
In interval rotation, the square bracket means that the end point is included. The round
bracket means that the end point is not included.
Graph
Inequality
Interval
a
b
axb
a, b
a
b
axb
a, b
a
b
axb
( a , b]
a
b
axb
[ a , b)
xa
( a , )
xa
[ a , )
xb
( , b)
xb
( , b]
   x   all reals
 , 
x  a or x  b
 , a   b, 
x  a or x  b
( , a ]  [ b, )
a
a
b
b
a
b
a
b
Periodic Functions
The values of sin  and cos  repeat themselves each 2 radians.







sin   4    sin   2    sin
6
6

6






cos   4    cos   2    cos
6
6

6

In general, sin  x  2  sin x , for any x.
In general, cos  x  2  cos x , for any x.
Functions with a repeating pattern are called periodic functions.
Mathematics C30
92
Lesson 2
y
y
y = f( x)
–2
2 4 6
y = g( x)
x
–8
–4
4
x
In y  f  x  the y values repeat themselves every 2 units. Therefore, f is a periodic function
and its period is of length 2 units.
Similarly, in y  g x  the y values repeat themselves every 4 units. Therefore, g is a
periodic function and its period is of length 4 units.
The sine and cosine functions are periodic, and each has a period of length 2 . If the part
of the graph between 0 and 2 is known, then the part of the graph from 2 to 4  is a
copy, etc.
Periodic Function
A function f is called periodic if there is some real number P such that
• f x  p  f x  for x in the domain of f.
• The smallest such number P is called the period of f.
The Sine Function
The equation, y  sin  , can be graphed using the unit circle.
Imagine the unit circle being cut at 0 or 2 and unrolled along the positive x-axis from 0
to 2 (6.28 = 2 ). The arc length becomes the distance along the x-axis starting from 0.
The graph can be drawn using the usual x-axis and y-axis as a frame of reference. The
y-axis will be labelled in the usual way, but the x-axis will be marked in radian units.
Mathematics C30
93
Lesson 2
y

2

4 
3

0
cu t
3
2
1
1
–2
–
–3
2
–
2

2
2
3

4 5
3
2
6 7
2

x
r adians
The diagram below shows that for any angle  the y-value in y  sin  is the same as the
ordinate ( y-value) of the point P on the unit circle.
y
1
P( x , y )
1

y

3
y

3
sin   sin

2
x

 y
 y
3 1
Also, note that the y-values increase from 0 to 1 as  increases from 0 to
Mathematics C30
94

.
2
Lesson 2
In this way the graph of y  sin  can be sketched over 0, 2  .
The y values:

•
decrease from 1 to 0 as  goes from
to  .
2
3
•
decrease from 0 to  1 as  goes from  to  .
2
3
•
increase from  1 to 0 as  goes from  to 2 .
2
One complete cycle of the sin curve is shown. The curve from 2 to 4  is a copy, etc.
y
1
y = sin 
0.5
–0.5

6

3

2
2

3
5

6
7

6

4

3
3

2
5

3
11 
6


–1
The portion of a periodic curve drawn over any one period is called one cycle of the curve.
Mathematics C30
95
Lesson 2
The Cosine Function
a

 a . Visualize, as  increases from 0 to , the a-value
2
1
decreases from 1 to 0. These a-values become the y-values on the graph of y  cos  .
For any angle  , cos  
( a, b)

6
1

a
y
1
0.5
–0.5
y = cos 

6

3

2
2

3
5

6

7
6
4

3
3

2
5

3
11 
6

2
–1
In this way one cycle of the graph can be sketched over 0, 2 .
The y-values:

•
decrease from 0 to  1 as  goes from
to  .
2
3
•
increase from  1 to 0 as  goes from  to
.
2
3
•
increase from 0 to 1 as  goes from
to 2 .
2
The curve from 2 to 4  is a copy, etc.
Mathematics C30
96
Lesson 2
The Tangent Function
For any angle , tan  
y
.
x

At   0 , tan   0 and as  increases to

As  continues to increase from

, tan  increases to 1.
4


to , the y-values increase, as the x-values decrease
4
2
y
to zero. Consequently, the ratio
increases indefinitely.
x
The vertical line  

is an asymptote to the curve.
2
P ( x, y )

y
Tan  
y

increases without bounds as   .
x
2



through the fourth quadrant, tan  starts at 0 and decreases
2

without bounds. The vertical line    is another asymptote to the curve.
2
As  goes from 0 to 
  
One full cycle of the curve is drawn over   ,  .
 2 2
Mathematics C30
97
Lesson 2
Asympt ote
Asympt ote
y
y  tan 
2
1

2
– 
2
3
2

2

4

–1
–2
Calculator Activity: Complete the table to get an indication of the graph of
 3 
y  tan  over the interval  ,   .
2 2 


 0 .01
2

 0 .02
2
3

4
5

4

3
  0 .02
2
3
  0 .01
2
3
  0 .001
2
tan 
Properties of the Sine, Cosine, and Tangent Curves
It is convenient to replace  with x when drawing graphs on the usual x, y-coordinate
plane.
Mathematics C30
98
Lesson 2
The following three sketches show one cycle of each of the graphs. Each will be
referred to as the principal cycle.
y
1

y  sin x
2
x
–1
y
1
y  cos x

x
2
–1
y
–

2

2
x
y  tan x
Each function is periodic. The period of sine and cosine is 2 (of length 2 ), and the
period of tangent is  (of length  ). The principal cycle is one sample copy of the graph
and the complete graph consists of copies of the principal cycles to the left and to the right.
The domain of any function is the set of x values for which a y-value exists. (x is the
independent variable and y is the dependent variable.) The range is the set of all y-values
for which there is a corresponding x value in the domain.
Mathematics C30
99
Lesson 2
Both sine and cosine have all real numbers as the domain and  1  y  1 as the range.
The tangent function has all real numbers y for the range. The domain is all real numbers
except those at which the asymptotes occur. The domain of y  tan x is all real numbers
 3 5

,
, ... . In general,  n  , n  I are the excluded from the domain of
except  , 
2
2
2
2
y  tan x .
The maximum value of a function is the greatest y-value in the range. The minimum
value of a function is the least y-value in the range. The tangent function has no
maximum or minimum values since the y-values increase or decrease indefinitely as they
approach the asymptotes.
Activity 2.4
State the x value at which each occurs.
In General
Maximum
in
Principal
Cycle
Minimum
in
Principal
Cycle
—
—
Zero in
Principal
Cycle
Maximum
Minimum
—
—
Zero
sin x
cos x
tan x
Mathematics C30
100
Lesson 2
The Cosecant Function
1
, the graph of the cosecant function can be constructed by finding the
sin x
reciprocal of each of the y-values of the sine function. An asymptote will occur for csc x
where sin x  0 . This will be at x = 0,  , and 2 since sine of these values is zero.
Since csc x 




 1 , since sin  1
2
2
3
3
csc
 1 , since sin
 1
2
2
csc
y = csc x
y
2
y = sin x
1
–1

2

3
2
x
2
One Cycle of the Cosecant Function
The Secant Function
1
, the graph of the secant function is constructed by finding the reciprocal
cos x
of each of the y-values of the cosine function. Asymptotes occur for sec x where cos x  0 .

3
This will be at x  and  since cosine of these values is zero.
2
2
Since sec x 
Mathematics C30
101
Lesson 2


sec 0  1 , since cos 0  1
sec   1 , since cos   1
y
y = sec x
2
y = cos x
1
–1


2
3
2
x
2
–2
One Cycle of the Secant Function
The Cotangent Function
1
, the graph of the cotangent function is constructed by finding the
tan x
reciprocal of each of the y values of the tangent function. Asymptotes occur for cot x
where tan x  0 and zeros for cot x occur where asymptotes of tan x occur.
Since cot x 




 1 , since tan  1
4
4
 
 
cot     1 , since tan     1
 4
 4
cot
y
2
1
– – 
2 –1

2

x
One cycle of the cotangent function is from 


to .
2
2
The principle cycle for sine, cosine, secant and cosecant is 0 to 2  .
The principal cycle for tangent and cotangent is
Mathematics C30
102


to .
2
2
Lesson 2
Exercise 2.4
1.
Explain why sin x  2 , and cos x  2  , are equal to sin x and cos x, respectively.
2.
Explain why sin x  2 or cos x  2 have no solution.
3.
For the cotangent function
a.
b.
c.
what is the period?
what is the domain?
what is the range?
4.
Sketch 3 cycles of the cotangent function with the asymptotes.
5.
For the cosecant function
a.
b.
c.
d.
what is the period?
what is the domain?
what is the range?
1
why does csc x  have no solution?
2
6.
Sketch 2 cycles of the cosecant function with asymptotes. Include the sine curve as
a guide.
7.
For the secant function
a.
b.
c.
d.
8.
what is the period?
what is the domain?
what is the range?
what is the solution to sec x  1 ?
Sketch 2 cycles of the secant function with the asymptotes. Include the cosine curve
as a guide.
Mathematics C30
103
Lesson 2
9.
Which of the graphs of the six trigonometric functions are symmetrical about the
y-axis? (The graph on one side of the y-axis is the mirror image of the graph on the
other side.)
10.
Which of the following are true?






sin  x   sin x
cos  x   cos x
tan  x   tan x
csc  x   csc  x 
sec  x   sec  x 
cot  x   cot  x 
(Hint: refer to the answer for number 10.)
11.
a.
The principal cycle of y  sin x begins at x = ________ and ends at
x = ________.
b.
The principal cycle of y  cos x begins at x = ________ and ends at
x = ________.
c.
The principal cycle of y  tan x begins at x = ________ and ends at
x = ________.
Mathematics C30
104
Lesson 2
2.5
Amplitudes, Period, and Phase Shift (Part 1)
There are many variations of the six trigonometric functions. The following example of
the location of a point on a ferris wheel as a function of time shows one of the variations.
A ferris wheel rotates once every minute, has a radius of 15 metres, and the lowest point is
5 metres above ground level. The height above the ground at any time t of a fixed point on
the edge of the wheel can be described by the periodic graph.
h
40
30
20
10
t
1
2
3
4
m in ut es
This curve does not have exactly the same properties as the curve of y  sin x , but
knowledge about the principal cycle of the sine curve can be used to describe the ferris
wheel curve.
In this section and in the next lesson, functions with equations of the following forms will
be graphed:
•
•
•
y  k  a sin bx  c
y  k  a cos bx  c
y  k  a tan  x  c
where a, b, c, and k are real numbers.
Mathematics C30
105
Lesson 2
Drawing graphs of trigonometric functions is simplified by the periodic nature of the
graphs. In many cases it is not always necessary to make a table of values and plot points.
You will learn how to sketch graphs of more complicated trigonometric functions by using
the properties and shape of the principal cycle of each of the functions, y  sin x , y  cos x ,
and y  tan x .
You will now learn how to sketch one cycle of each of the three trigonometric functions if
1

the x is replaced by something more complicated such as y  sin 2 x   , y  cos  x    ,
3


or y  tan  x  2 .
Example 1
Sketch one cycle of the graph of y  sin 2 x   .
Solution:

This graph will have the same sinosoidal (wavy) shape as the graph of y  sin x , except
that this graph will begin and end at different positions on the x-axis.
Determine where the graph of one cycle begins and ends.
y  sin x
Principal cycle
begins when x = 0
ends when x  2
By comparing with the above rectangle, we have the following.
y  sin 2 x  
One cycle
begins when 2 x    0
Mathematics C30
ends when 2x    2
106
Lesson 2
Solve for x:
2x    0
2x  

x
2
2 x    2
2 x  2  
2 x  3
3
x 
2
Cycle Begins

Cycle Ends
Therefore, one cycle of the graph of y  sin 2 x   begins at x 

3
and ends at x   .
2
2
Determine the midpoint of one cycle.
 3 4 


3
2
2  2  2   .
The midpoint between
and  is
2
2
2
2
2
Use this information and the knowledge of the general shape of one cycle of a sine curve to
sketch the graph.
Label the points on the x-axis and sketch the shape of the sine curve.
y
y = sin (2 x –  )
1
–1
Mathematics C30

2
5

4
3
4
107

3  2
2
x
Lesson 2
Example 2
1

Sketch one cycle of the graph of y  cos  x    .
3

Solution:
Determine where the graph of one cycle begins and ends.


1

Since the principal cycle for y  cos x begins at x = 0, the cycle for y  cos  x   
3

1

begins when  x     0 .
3

1
x0
3
1
x
3
x  3
Begins
1

Since the principal cycle for y = cos x ends at x  2 , the cycle for y  cos  x    ends
3


1

at  x     2  .
3

1
x    2
3
1
x  3
3
x  9

Ends
1

Therefore, one cycle of the graph of y  cos  x    begins at x  3 and ends at
3

x  9 .
Determine the midpoint of one cycle.
The midpoint between 3 and 9 is
Mathematics C30
3   9  12 

 6 .
2
2
108
Lesson 2
Label the x-axis and sketch the shape of the cosine curve.
y
1

y  cos  x   
3

1



x
–1
Example 3
2 

Sketch one cycle of the graph of y  tan 3 x    .
3 

Solution:
Determine where the graph of one cycle begins and ends.


2 

Use the distributive property to change y  tan 3 x 
 to y  tan 3 x  2 .
3 


The principal cycle for y  tan x begins at x   . Therefore, the cycle for
2

2 

y  tan 3 x    begins at 3 x  2    .
2
3 

3 x  2  

2

4 
5

 
2
2 2
2
1  5 
x  

3 2 
5
x
Begins
6
3 x  2  
Mathematics C30
109
Lesson 2

Similarly, since the principal cycle of y  tan x ends at x 
y  tan 3 x  2 ends at 3 x  2  

.
2

, the cycle of
2

2

 4
3
3 x   2  

2
2 2
2
1  3 

x  

3 2 
2
3 x  2 

Therefore, one cycle of the graph of y  tan 3 x  2 begins at x  
5

and ends at  .
6
2
Draw vertical asymptotes at the beginning and at the end of the cycle, and then sketch the
shape of the tangent curve.
y = t an(3x + 2)
y
–5
6
Mathematics C30
– 4
6
–
3
6
x
110
Lesson 2
Activity 2.5
This activity is to be submitted with Assignment 2.
Use the principal cycle of each trigonometric function to sketch the graph of each
equation.
Cycle Begins at
Cycle Ends at
Sketch
yy
1. a.
y  sin 2 x
xx
b.
y
2 

y  sin  2 x 

3 

x
y
c.


y  sin  2 x  
3

Mathematics C30
x
111
Lesson 2
y
2. a.
y  cos 3 x
x
y
b.
y  cos 2 x  
x
y
c.


y  cos  2 x  
2

x
y
3. a.
y  t an 2 x
x
Mathematics C30
112
Lesson 2
y
b.


y  tan  x  
3

x
y
c.


y  tan  x  
3

x
Period, Phase Shift, and Amplitude
The period of a trigonometric function is the length of one cycle.
Period = |End of Cycle – Beginning of Cycle|

The absolute value insures a positive period.
The period can be determined from the preceding examples.



The period of y  sin 2 x   is
3
 2
 
 .
2
2
2
1

The period of y  cos  x    is 9   3  6  .
3

  5 
 5 
 .
The period of y  tan 3 x  2 is    
 
2  6 
2 6
3
Mathematics C30
113
Lesson 2
The phase shift of the trigonometric functions sine and cosine is the amount of change
from the beginning of the principal cycle to the beginning of the cycle of the given function.



The graph of y  sin     illustrates a phase shift of  from the principal cycle.
4
4



y  sin    
4

y  sin x
The cycle starts at 0
The cycle starts at 
and ends at 2 .
and ends at

4
7
.
4
Phase Shift is determined by the shift at the beginning of the cycle.
The end point of the cycle plays no role in phase shift.

.
2

The phase shift of y  sin 2 x   is

1

The phase shift of y  cos  x    is 3  .
3

A negative phase shift means that the beginning of the cycle has been shifted to the left,
and a positive phase shift means that the beginning of the cycle has been shifted to the
right from the principal cycle.
Phase shift for the tangent function can be defined similarly as the amount of shift from

the beginning of the cycle  .
2
Mathematics C30
114
Lesson 2
For example, the cycle of y  tan  x  2 begins at 
5
 . The phase shift can be
2
5

determined by finding the difference between   and  .
2
2


5
5

 
       
2
2
2
 2
4

2
 2 
The phase shift is  2  , or 2  units to the left of 

.
2
Amplitude
When a periodic function has a maximum value M and a minimum value m, the function
M m
has an amplitude of
.
2
For example, the graphs of y  sin x and y  cos x each have a maximum of M  1 and a
M m
mimimum of m  1 . The amplitude can be determined by solving
.
2
M  m 1   1  1  1 2


 1.
2
2
2
2
This represents the distance from the maximum or the minimum of the graph to the
“middle” of the graph.
y
y
M
M
x
m
m
Mathematics C30
115
x
Lesson 2
The coefficient of the trigonometric function is important in determining the amplitude.


y  3 cos x
y  1 .5 sin x
The amplitude is 3.
The amplitude is 1.5.
The amplitude of the graph is exactly the absolute value of the coefficient of the
trigonometric function.
To explain this, note that if a is positive in y  a sin  , then the maximum is a 1  a and
the minimum is a   1  a .
M  m a   a  a  a


a.
2
2
2
Note the difference between a positive value for a and a negative value for a.
The amplitude is then
y  1 sin x
y  1 sin x
Amplitude is not defined for the tangent, cotangent, secant or cosecant functions
since the graphs have no maximum or minimum.
Vertical Translations of Trigonometric Functions
If a constant is added to a trigonometric function, then the graph of the function is raised
or lowered by the amount of that constant.
The following diagrams illustrate vertical translations of the principal cycle of y  sin x .
y
y
1
y
3
x
2
1
x
–1
1
1
x
–3
–1
y  sin x
(principal cycle in
standard position)
Mathematics C30
–2
y  2  sin x
116
y  2  sin x
Lesson 2
Although each of the graphs are at different distances from the x-axis, the amplitude of
each of the functions is the same as that of the principal cycle in standard position.

For y  2  sin x , amplitude

M  m  1   3   1  3


1.
2
2
2
Note the difference between y  cos  x  2  and y  cos x  2 .



y  cos x  2 really means y  2 + cos x .
y  cos  x  2  cannot be changed.
It is best to write y  2  cos x rather than y  cos x  2 to avoid confusion.
Exercise 2.5
1.
Determine the amplitude, period, and phase shift for each of the following equations.
y  3 sin
c.
y  tan 2 x
1


cos  3 x  
3
4



y  3 tan  x  
2

y
e.
g.
2.
1
x
3
a.
b.
y  cos 5 x
d.

1
y  3 sin  x  
2
3
f.
y  5 tan  x  
State the amplitude, period, and phase shift for each of the following graphs.
a.
b.
y
y
6
3

x
2
–
10
–6
x
–3
cosine
sine
Mathematics C30


117
Lesson 2
c.
d.
y
y
3
1




x
x
–3
1
of a sine cycle is shown.
2
3.
–
1
of a cosine cycle is shown.
2
Determine the maximum and minimum values.
a.
b.
y  2 sin x  3
1
y   2 sin 4 x  
2
Mathematics C30
118
Lesson 2
Answers to Exercises
Exercise 2.1 1.
y
{ 105°, 7  /12}
{ 120°, 2  /3}
{ 90°,  /2}
{ 75°, 5 /12}
{ 60°,  /3}
{ 135°, 3 /4}
{ 45°,  /4}
{ 150°, 5 /6}
{ 30°,  /6}
{ 15°,  /12}
{ 165°, 11 /12}
{ 0°, 0} &
{ 360°, 2 }
{ 180°,  }
{ 195°, 13 /12}
x
{ 345°, 23  /12}
{ 330°, 11 /6}
{ 210°, 7 /6}
{ 225°, 5 /4}
{ 315°, 7 /4}
{ 300°, 5 /3}
{ 240°, 4 /3}
{ 255°, 17 /12} { 270°, 3  /2}{ 285°, 19  /12}
2.

6
3.

4
Exercise 2.2 1.
a.
7.2446
f.
0.4200
b.
0.2493
g.
0.6101
c.
0.1736
h.
1.0472
d.
1.7782
i.
0.2450
e.
 0.1606
j.
0.2450
Mathematics C30
119
Lesson 2
2.
c.
d.
1
1
, 1, , 0
2
2
2 3
2, 1,
, undefined
3
 1 , 0, undefined, 0
 1 , undefined,  1 , 0
a.
sin
a.
b.
3.
b.
c.
d.
4.
a.
b.
c.
d.
5.
a.
b.
c.
d.
Mathematics C30

3

3
2

3
 tan  
6
3

2
 cos  
4
2

2 3
 sec  
6
3


3

 sin  
4

 cos  
6

tan  1
4
 sin
e.
 cos
f.
csc
g.
3
2
2
2
3
2
0
e.
 sec
g.
h.

 3
3

2
 sin  
4
2

3
cos 
6
2
 1
cos 
3 2
 tan
e.
f.
g.
h.
120

2
6

 cot  1
4
h.
f.

1

3
2

 2
3

 csc  2
6

3
tan 
6
3
0


3

2
cos 
4
2

 cot  
6

 csc  
4
 sin
3
2
3
2
Lesson 2
6.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
7.

8.
a.
b.
9.
a.
b.
c.
d.
Mathematics C30

2

0, 
0, 
 3
, 
2 2
0, 
 5
, 
6 6
2
4
, 
3
3
3
5
, 
4
4
4
5
, 
3
3

2
5
 , cos    cos  
4
2
4
1
1
1  1  1 1

   1
2 2
2
2 2 2
 2 2    1   1   4 1
4
 2  2 

(0, 1)
(0, 1)


 3 , 1 
 2 2


 1

 , 3 
 2 2 


121
Lesson 2
Exercise 2.3 1.
a.
s  r
27 cm  r
r
5
4
4  27
 6.88 cm
5
b.
10.47 cm
c.
2
d.
1.60 m
2.
 15 
(100 cm) 
 = 26 cm
 180 
3.
v  ra
 2 
(6400 km) 
 = 1676 km/h
 24 h 
4.
5.
Mathematics C30
2 000 000 cm/h = (35 cm)a
a  57142 .857 radians/h
 15 .87 radians/s
15 .87

 2 .5 rev/s
2
2 2  radians
11 s

4
/ s  1 .14 /s
11
122
Lesson 2
6.
588 m
7.
a.
b.
c.
Exercise 2.4 1.
120
 2 /s
60
120 × 60 = 7200/h
120 60

rev/min
2

Both functions are periodic with period 2 and repeat every 2 units.
2.
The range of both functions is  1  y  1 .
3.
a.
b.
c.

all real numbers except n , n  I
all real numbers
4.
y
– – 
2
5.
a.
b.
c.
d.
Mathematics C30

2

3
2 

x
2
all real numbers except n , n  I
(,  1]  [1, )
This means y  1 or y  1 .
1
is outside the range of the function
2
123
Lesson 2
6.
y
1
–2
7.
–
3

2
–
–

2
–1

2
a.
2
b.
all real numbers except
c.
d.
(,  1]  [1, )
x    n2, n  I

3

2

2

3
 n2 and   n2 , n  I
2
2
8.
y
1
– 2
3
–

2
–

2

–
2

x
3

2
2
–1
Mathematics C30
124
Lesson 2
9.
y  cos x and y  sec x
10.
cos  x   cos x
sec  x   sec x
11.
a.
b.
c.
0, 2
0, 2
 
 ,
2 2
Exercise 2.5
1.
2.
Mathematics C30
6
2

5

2
a.
3,
b.
1,
c.
undefined
d.
3
6
e.
1
3
2
3
f.
undefined

g.
undefined

a.
6,
b.
3,
c.
1,
d.
3,
8 ,
5
,
2
3
,
2
4,
0
0
   
   
4  2 4
3

2

12
3   

     
2  2
  
0    
 2 2

2


4

4

125
Lesson 2
3.
a.
maximum:
minimum:
b.
maximum:
minimum:
Mathematics C30
23  5
23 1
1
1
2  2
2
2
1
1
 2  1
2
2
126
Lesson 2
Mathematics C30
Module 1
Assignment 2
Mathematics C30
127
Lesson 2
Mathematics C30
128
Lesson 2
Optional insert: Assignment #2 frontal sheet here.
Mathematics C30
129
Lesson 2
Mathematics C30
130
Lesson 2
Assignment 2
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
check () beside it.
1.
2.
3.
The arc of a circle of radius 1 m is also the length of 1 m. The central
angle of the arc has measure ***.
____
____
a.
b.
____
c.
____
d.
A 76° angle has a radian measure of approximately ***.
____
a.
____
____
____
b.
c.
d.

180
76
4354
1.326
An angle of measure 
____
____
____
____
Mathematics C30
57.3°
1°

4
2
a.
b.
c.
d.
4
 is in quadrant ***.
3
1
2
3
4
131
Lesson 2
4.
5.
An angle that is a multiple of 90° is ***.
____
a.
____
b.
____
c.
____
d.

6
7

2
11

4

12
An angle with measure
point ***.
6.
Mathematics C30
____
____
a.
b.
____
c.
____
d.
7
 has a terminal ray which passes through the
4
 1, 1
1,  1
1, 2 
 1, 2 
The reference angle of an angle whose measure is
8
 is ***.
3

3
____
a.

____
b.
____
c.
____
d.
5

3
7

3

3
132
Lesson 2
7.
8.
9.
The expression tan
____
a.
____
b.
____
c.
____
d.

6

tan
6
7
 tan 
6
6
cot 
7
 tan
9 
The expression csc    is equivalent to ***.
4 

4
____
a.
csc
____
b.
 csc
____
c.
____
d.

4

 sin
4
1
9 
sin   
4 
A ferris wheel of radius 10 m makes 15 revolutions for one ride. The
circular distance traveled by a rider is ***.
____
____
____
____
Mathematics C30
7
 is equivalent to ***.
6
a.
b.
c.
d.
20 m
100 m
200 m
942 m
133
Lesson 2
10.
The number of revolutions made by a bicycle wheel with a 0.15 m radius
in traveling 1000 m is ***.
____
____
____
____
11.
12.
____
____
____
a.
b.
c.
____
d.
 , 
0, 
0, 2 
 
 ,
2 2
The range of the function y  5 cos x is ***.
a.
b.
c.
d.
 5, 5
2
0, 2
 1, 1
The maximum value of y in the graph y  10  3 sin x is ***.
____
____
____
____
Mathematics C30
3333
530
1061
6666
The principal cycle of the function y  tan x begins and ends,
respectively, at ***.
____
____
____
____
13.
a.
b.
c.
d.
a.
b.
c.
d.
3
7
 10
 13
134
Lesson 2
14.
15.
One cycle of the function y  sec x is over the interval ***.
____
a.
____
____
b.
c.
____
d.
  3 
 2 , 2 
0, 
0, 2
  
 2 , 2 
The value of sin
____
____
____
____
a.
b.
c.
d.
 1
 is ***.
6 2
0.5
1
1.0236
1.5
16. The range of y  sec x is ***.
____
____
____
____
a.
b.
c.
d.
all real numbers
0, 2
 1, 1
( ,  1]  [1, )
17. The principal cycle of y 
Mathematics C30
____
a.
____
b.
____
____
c.
d.
3
1

tan  x    begins at ***.
4
2

 3


2
 2
0
135
Lesson 2
18. The amplitude of y 
____
a.
____
b.
____
c.
____
d.
5
tan 2 x   is ***.
6

2
2
5
6
not defined
19. The principal cycle of y  2 
____
a.
____
b.
____
c.
____
d.
1
cos 6 x  2  ends at ***.
2

3

2

3
2
20. The expression 3  sin 2 x  4 is the same as ***.
____
____
____
____
Mathematics C30
a.
b.
c.
d.
3  sin 2 x  4 
3  sin 2 x  2 
7  sin 2 x
sin 2 x  7 
136
Lesson 2
Answer Part B and Part C in the space provided. Evaluation of your solution
to each problem will be based on the following.
(5)
(4)
B.
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
3
, find the principal values of  and evaluate the exact
2
value of sin  and tan  .
1.
If cos   
2.
Express each in terms of the reference angle, and give the exact value.
Mathematics C30
a.
 7 
sin    
 3 
b.
 5 
cos  

 4 
137
Lesson 2
c.
d.
(6)
3
tan
11

6
 
csc   
 4
2
, sketch the possible principal angles and find the exact
5
values of sec A and csc A.
If cot A  
y
x
Mathematics C30
138
Lesson 2
(6)
(5)
4.
5.
Mathematics C30
On a circular, 2 km race track, a car has reached a speed of 200 km/h.
(Refer to Section 2.3)
a.
Find the radius of the track in kilometers.
b.
Find the speed of the car in km/min.
c.
Find the angular speed in radians per minute.

1
Sketch the principal cycle of the curve of y  1  sin  x   .
4
2
Label the maximum and minimum points, and show the calculations
used to determine the beginning and end of the cycle.
139
Lesson 2
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6.
Mathematics C30
State the amplitude, and calculate the phase shift and period for each
function. State the maximum value of y if applicable.
a.
y  4 cos 3 x  
b.
y  4 sin 3 x  
c.

1
y  5  7 cos  x  
4
3
140
Lesson 2
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7.
Mathematics C30
d.
4 
1
y  5  8 sin  x   
3 
3
e.
y  5 tan  x  
Submit Activity 2.5
141
Lesson 2
(10)
C.
1.
At the end of a spring, a ball rises and falls about its rest position. The
distance, d (in centimetres), is given by the equation
d  4 cos 12 t  , where t is the time in seconds.
m axim um
d
r est
d
Mathematics C30
m in im um
a.
From the equation, determine the amplitude, period and phase
shift.
b.
Draw the graph of the function.
c.
For what values of t does the ball return to its original position?
d.
For what values of t is the ball at its rest position?
e.
For what values of t is the ball at its maximum height?
142
Lesson 2
(5)
2.
Write a point form summary (no more than 2 pages in length) of this
lesson on a separate page. In this summary include definition
formulas, technique and examples which should help you review for
exams.
_____
(100)
Mathematics C30
143
Lesson 2
Mathematics C30
144
Lesson 2