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Chapter 4 Trigonometric Functions
4.1
4.1.1
Angles in Standard Position - Definitions
An angle is formed by rotating an initial arm about a fixed point.
An angle is said to be in standard position when the initial arm
is on the positive x-axis and the vertex is at (0, 0).

Positive angles have a
counterclockwise rotation.
Negative angles have a
clockwise rotation.
A principal angle is the angle measured from the positive x-axis
to the terminal arm. The principal angle is always a positive
measure.
A reference angle is the angle measured from the closest x-axis
to the terminal arm. The reference angle is always an acute angle
and is always positive.
Coterminal angles are angles that share the same terminal arm.
4.1.2
Sketching Angles in Standard Position
Sketch the following angles in standard position.
State the principal angle, the reference angle, and
one positive and one negative coterminal angle.
a) 1500
b) -2600
c) 5600
Principal Angle
1500
Principal Angle
1000
Principal Angle
2000
Reference Angle
300
Reference Angle
800
Reference Angle
200
Coterminal Angles 5100
-2100
Coterminal Angles 4600
-6200
Coterminal Angles 5600
-1600
4.1.3
Sketching Angles in Standard Position
d) -2200
Principal Angle
1400
Reference Angle
400
Coterminal Angles
5000
-5800
To find all coterminal angles:
 = 140 + 360n where n is an
element of the integers.
4.1.4
Radian Measure
A radian is the measure of the angle at the centre of the circle
subtended by an arc equal in length to the radius of the circle.
   1 rad
r

   2 rad
r

r
r
2r
r
arc length
number of radians =
radius

a
r
4.1.5
Changing Degree Measure to Radian Measure
r
   1 revolution of 360
The arc length is the cirumference, 2 r .
a

r
2r

r
 = 2
Therefore, 2 rad = 3600.
Or,  rad = 1800.
4.1.6
Changing Degree Measure to Radian Measure
Calculate the radian measure:
a) 2100
b) 3150
1800 =  rad

1 
180

210 
 210
180
7

6
Exact radians
= 3.67
Approximate radians
315 

180
 315
7  radians

4
4.1.7
Changing Radian Measure to Degree Measure
Calculate the degree measure:
a) 2 rad
3
 rad = 1800
180
1 rad 

2 180 2


3

3
= 1200
c) 1.68 rad
180
1.68 rad 
 1.68

= 96.260
b)  rad
12

180 
rad 

12

12
= 150
To convert from radians
to degrees, multiply by
180
.

To convert from degrees
to radians, multiply by

.
180
4.1.8
Finding the Sector Angle or the Arc Length
Find the measure of the sector angle:
5 cm

6.1 cm
a
 =
r
6.1
 =
5
  1.22
Find the arc length:
8 cm
700
4.1.9
Convert 700 to radians:
a

7
 70 

180 18
The arc length is 9.77 cm.
a
 =
r
7 a

18 8
18a = 56
a = 9.77
Angular Velocity
Radians are often involved in applications involving angular speed.
Angular speed is the rate at which the central angle is changing.
Find the average angular speed of a
wheel that is rotating 15 times in 3 s.
Each time that the wheel rotates, it
turns through a central angle of 2 radians.
Therefore, in 15 rotations, the wheel has travelled 30 radians.
dis tan ce
speed 
time
speed = 30
3
speed = 10 rad/s
The average angular speed
is 10 rad/s.
4.1.10