1. ANGLES
and
THEIR
MEASURES
Trigonometry 002

2. Outline:
Basic Terms
The Degree Measure
The Revolution Measure
The Radian Measure
• Length of an Arc
• Integral Multiples of Special Angles

4. B
A
O
Figure 1

5. 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.

6. Counterclockwise Rotation
Clockwise RotationPositive Mesures
Negative Mesures
initial side
x
y
initial side
x
y

7. The Degree Measure

11. 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
B
O
A
B
O
A
O
B

12. 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.

13. Consider the following:
r
r
A
B
O
Figure 3.a

14. Figure 3.b
1
A
B
O
1
1

15. Figure 3.c
A
B
O
2
2

16. A
B
O

17. Consider the following example.

18. Degrees Revolutions Radians
1 rev

19. Let us try this...

20. Degrees Radians Revolutions
0 0 0
60
120
180
240
300
360

21. Degrees Radians Revolutions
0 0 0
45
90
135
180
225
270
315
360

22. Degrees Radians Revolutions
0 0 0
30
90
150
180
210
270
330
360

23. Try this one:
Find the measure of each angle in degrees, radians, and revolutions.
2.625 rev

26. 2.625 rev

27. 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

28. Exercises:

29. 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