7-1 Measurement of Angles

Definitions
  1)  Angle - two rays joined at a common point called a vertex point.
 
Demo: Ferris Wheel (Manipula Math)
 
  2)  Revolution - a common unit used to measure large angles, like the number of revolutions a car wheel makes traveling at 10 mph.
 
  3)  Degree - a common unit used to measure smaller angles.   There are 360 degrees in 1 revolution.  1/2 of a revolution = 180 degrees, 1/4 rev = 90o
 Degrees can be divided into smaller units of minutes and seconds.  1 degree equals 60 minutes, while 1 minute equals 60 seconds.
Examples
15.4o = 15o + .4(60)' = 15o 24'
50o30''15" = 50o + (30/60)o + (15/3600)o = 50.5042o
  4)  Radian - the measure of a the central angle when an arc of a circle has the same length as the radius of the circle.
 
 
  5)  Radian measure - the number of radius units in the length of an arc AB
s = r0

Changing radians to degrees and degrees to radians.
 
To change degrees to radians, multiply by p/180
310o = 310 x p /180 = 31 p/18 rads
 
To change radians to degrees, multiply by 180/p
 
3p = 3px 180/p = 540o
5 rads = 5 x 180/p =286.5o

Angles in the co-ordinate system
        An angle in the co-ordinate system is usually placed in standard position.  This means that the vertex is at the origin and its initial ray is along the positive x-axis.  A counterclockwise rotation is considered to be positive and a clockwise rotation is considered to be negative.  If the terminal side of an angle is standard position lies along an axis, the angle is said to be a qadranutal angle.  Two angles in standard postion are called coterminal if they have the same terminal side.

Samples
1)  Find two angles with the same terminal side, one positive and one negative for each angle.
        a)  120o
                        Add 360 to find another positive  120 + 360 = 480
                        Subtract 360 to find a negative   120 - 360 = -240o
 
        b)  400o
                        Add or subtract 360 for a positive.  400 - 360 = 40o
                        Subtract enough 360's to make it negative.  400 - 360 - 360 =
-320o