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ANGLES and
THEIR MEASURES
Trigonometry 002
Outline:
Basic Terms
The Degree Measure
The Revolution Measure
The Radian Measure
• Length of an Arc
• Integral Multiples of Special Angles
In trigonometry, an angle is generated by a ray
rotating about a point. One side of the angle rotates
about a common endpoint and the other side remains
stationary. The stationary ray is the initial side of the
angle, and the rotating ray is the terminal side.
Angles may be negative or may measure greater
than .
In plane geometry, an angle is defined as a
figure formed when two rays meet at a single point
called the vertex. Furthermore, angle measures are
limited to values between and .
B
A
O
Figure 1
In figure 1, is a result
of the rotation of side to side
about point O. Side is the
initial side and side is the
terminal side.
The measure of an angle is a number that
indicates the size and direction of the rotation which
forms the angle.
When the angle is rotated counterclockwise
direction it is given a positive sign; negative, if rotated
clockwise direction.
Counterclockwise Rotation
Terminal side
Clockwise RotationPositive Mesures
Negative Mesures
initial side
Term
inal
sid
e
x
y
initial sidex
y
The Degree Measure
An angle may be measured in degrees where
1 degree = 60 minutes = 3 600 seconds
1 minute = 60 seconds
1 complete revolution =
Example:
1. Convert in degrees form.
2. Convert in degrees, minutes, seconds form.
Solution:
1. Convert in degrees form.
=
=
=
Solution:
2. Convert in degrees, minutes, seconds.
= =
=
=
=
Try this one:
1. Convert in degrees form.
2. Convert in degrees, minutes, seconds form.
Answer:
1.
2.
The Revolution Measure
Since trigonomteric angles involve rotations of the
terminal ray, angles may be measured in terms of the number
of rotations or part of it. One complete rotation is called a
revolution.
A
BO𝜶
A
B
O
𝜽
A
O
B 𝜷
1 revolutioncounterclockwise
½ revolutionclockwise
revolutioncounterclockwise
The Radian Measure
The radian measure is based on the central angle of
a circle, its intercepted arc, and its radius.
The radian measure of a central angle is the number
of radius units in the length of the arc intercepted by the
angle.
One radian is the measure of a central angle of a
circle that intercepts an arc whose length is equal to the
radius of the circle.
Consider the following:
r
r𝜃 A
B
O
Figure 3.a
In figure 3.a,
Figure 3.b
In figure 3.b,
For central angle , the radius is 1, while the arc length AB is 3, so
1
𝒓=𝟏
A
BO
1
1
Figure 3.c
In figure 3.c,
For central angle , the radius is 2, while the arc length of AB is 4 clockwise, hence
𝒓=𝟐
A
B
O2
2
𝒓
A
B
O 𝜃𝒔
For central angle , whose intercepted arc AB has length s and radius OA has length r, the measure in radians is
Thus if the radius r of a circle and a central angle in radians are given, the intercepted arc has length
Consider the following example.
GIVEN: A circle with radius r and a central angle in radians intercepting an arc of length s.
a. If and , find .
b. If radians and , find .
c. If and , find .
ANSWER:a. radiansb. c.
Degrees Revolutions Radians
rev rad
rev rad
rev rad
1 rev rad
Let us try this...
1. Find the radian measure for the special angles , , and .
2. Compute the number of revolutions for each angle in (1).
Round off decimals to three decimal places\
3. Convert the following:
a. radians to degrees
b. 2.35 radians to degrees, minutes, seconds. (use )
Degrees Radians Revolutions
0 0 0
60
120
180
240
300
360
𝟓𝝅𝟑
𝟒𝝅𝟑
𝝅
𝟐𝝅𝟑
𝝅𝟑
𝟎
Integral Multiples of or
Degrees Radians Revolutions
0 0 0
45
90
135
180
225
270
315
360
𝝅 𝟎
Integral Multiples of or
𝟕𝝅𝟒
𝟓𝝅𝟒
𝟑𝝅𝟒
𝝅𝟒
𝝅𝟐
𝟑𝝅𝟐
Degrees Radians Revolutions
0 0 0
30
90
150
180
210
270
330
360
𝝅
Integral Multiples of or
𝝅𝟏𝟏𝝅𝟔
𝟕𝝅𝟔
𝟓𝝅𝟔
𝝅𝟔
𝟎
𝝅𝟐
𝟑𝝅𝟐
𝟓𝝅𝟑
𝟒𝝅𝟑
𝟐𝝅𝟑
𝝅𝟑
𝟎
Try this one:
Find the measure of each angle in degrees, radians, and revolutions.
𝜶
rev
𝑨𝑶
𝑷
𝜸 𝑨
𝑷
𝑶2.625 rev
𝑨
𝑷
𝑶𝜷
rev
The terminal side is rotated counterclockwise
rev from the initial side.
𝜶
rev
𝑨𝑶
𝑷
In radians:
In degrees: rev
In revolution units: rev
The terminal side is rotated
clockwise 1 rev plus rev
from the initial side.
In revolution units:
In degrees:
In revolution: rev
𝑨
𝑷
𝑶𝜷
rev
The terminal side is rotated
counterclockwise 2 rev plus rev
from the initial side.
In radians:
In degrees:
In revolution: rev
𝜸 𝑨
𝑷
𝑶2.625 rev
Quiz Time When solving problems, it is not unusual
to work on a solution for some time only to find out in the end that is wrong and you have to start all over. Failure is an opportunity to begin again intelligently.
-Henry Ford
Exercises:
Convert each angle measure to (a) degrees and (b)
radians and be able to illustrate it also.
1. 3.45 rev, clockwise
2. rev, counterclockwise
3. rev, counterclockwise
4. rev, clockwise
5. rev, counterclockwise
Thank You!!!
A correct understanding of the main formal sciences, logic, and mathematics is the proper and only safe foundation for a scientific education. - Arthur Lefevre