E N D
1. Angles and Their Measure
2. Angles
3. Angles of the Rectangular Coordinate System
4. Measuring Angles Using Degrees
5. Coterminal Angles An angle of x is coterminal with angles of
x + k 360
where k is an integer.
6. Text Example
7. Text Example cont.
8. Finding Complements and Supplements For an x angle, the complement is a 90 x angle. Thus, the complements measure is found by subtracting the angles measure from 90.
For an x angle, the supplement is a 180 x angle. Thus, the supplements measure is found by subtracting the angles measure from 180.
9. Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.
10. Radian Measure Consider an arc of length s on a circle or radius r.
The measure of the central angle that intercepts
the arc is
? = s/r radians.
11. Conversion between Degrees and Radians Using the basic relationship ? radians = 180,
To convert degrees to radians, multiply degrees by (? radians) / 180?
To convert radians to degrees, multiply radians by 180? / (? radians)
12. Example Convert each angle in degrees to radians
40
75
-160
13. Example cont. Solution:
40 = 40*?/180 = 2 ? /9
75 = 75* ? /180 = 5 ? /12
-160 = -160* ? /180 = -8 ? /9
14. Length of a Circular Arc Let r be the radius of a circle and ? the non-
negative radian measure of a central angle
of the circle. The length of the arc
intercepted by the central angle is
s = r ?
15. Example A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2?/3
Solution:
s = (7 inches)*(2 ? /3) =14 ? /3 inches
16. Definitions of Linear and Angular Speed If a point is in motion on a circle of radius r through an angle of ? radians in time t, then its linear speed is
v = s/t
where s is the arc length given by s = r ?, and its angular speed is
? = ?/t
17. Linear Speed in Terms of Angular Speed The linear speed, v, of a point a distance r from the center of rotation is given by v=r? where ? is the angular speed in radians per unit of time.
18. Angles and Their Measure