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# Angles and Measures

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### Angles and Measures

1. 1. ANGLES and THEIR MEASURES Trigonometry 002
2. 2. Outline: Basic Terms The Degree Measure The Revolution Measure The Radian Measure • Length of an Arc • Integral Multiples of Special Angles
3. 3. B A O Figure 1
4. 4. 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.
5. 5. Counterclockwise Rotation Clockwise RotationPositive Mesures Negative Mesures initial side x y initial side x y
6. 6. The Degree Measure
7. 7. 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
8. 8. 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.
9. 9. Consider the following: r r A B O Figure 3.a
10. 10. Figure 3.b 1 A B O 1 1
11. 11. Figure 3.c A B O 2 2
12. 12. A B O
13. 13. Consider the following example.
14. 14. Degrees Revolutions Radians 1 rev
15. 15. Let us try this...
16. 16. Degrees Radians Revolutions 0 0 0 60 120 180 240 300 360
17. 17. Degrees Radians Revolutions 0 0 0 45 90 135 180 225 270 315 360
18. 18. Degrees Radians Revolutions 0 0 0 30 90 150 180 210 270 330 360
19. 19. Try this one: Find the measure of each angle in degrees, radians, and revolutions. 2.625 rev
20. 20. 2.625 rev
21. 21. 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
22. 22. Exercises:
23. 23. 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