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