Measurement of Angles

An angle represents a rotation about a point.  Angles are drawn on the Cartesian Rectangular Coordinate System by placing the vertex at the origin and the initial side on the positive x-axis.  Include a little arrow to indicate the direction of rotation.  This is called drawing an angle in standard position.

Positive angles make counterclockwise rotations (the arrow goes up).

Negative angles make clockwise rotations (the arrow goes down).

Angles may be measured in revolutions, in degrees-minutes-seconds, and in radians.

  1. Revolutions are the number of times the angle makes a complete circle.
  2. 1o = 60' (One degree = 60 minutes)

    3. 1' = 60" (One minute = 60 seconds)
    1. To convert 34.159o to degrees-minutes-seconds
      1. Subtract 34 from the number.  There are definitely 34o.
      2. Multiply .159 by 60 to see how many minutes are included in this decimal of a degree.  The answer is 9.54'.
      3. Subtrace 9 from this number.  There are definitely 9'.
      4. Multiply .54 by 60 to see how many seconds are included in this decimal of a minute.  The answer is 34.40 seconds.  This number may be rounded to the nearest whole second since this is close as we are going to measure.
      5. Answer:  34o9'32"
    2. To convert 34o9'32" back to decimal degrees
      1. Remember, there are 60 minutes in each degree and 60 seconds in each minute, so there must be (60)(60) or 3600 seconds in each degree.
      2. Now, simply compute 34 + 9/60 + 32/3600 to get 34.159 back.
  3. A radian is the measure of an angle when an arc length equal in length to the radius is cut out by that angle.  The number of radii in the arc length is the radian measure.

    4. This means 0 = s/r.

    In this case, s = p and r = 3, soq = p/3
     
     

    1. Since C = 2pr and 0 = s/r, then 1 revolution's circumference = 2pr

      2. = 2p radians.                                                                             r
    2. So 360o = 2p radians
    3. And 1o = 2p radians.  =    p   degrees

      4.              360                    180
      1. Convert 220o to radians:  220p  =  11p

        2.                                         180          9
    4. And 1 radian = 360  = 180

      5.                         2p        p
      1. Convert 15 radians to degrees: 15 (180)  =  859.437o

        2.                                                                     p
Quadrantal Angles are angles whose sides lie on an axis.

First Quadrant angles are angles whose terminal sides lie in quadrant I.

Coterminal Angles are angles that share the same two sides.

  1. Find an angle coterminal with p/6.

    2. p/6 + 2p = 13p/6 or p/6 - 2p = -11p/6
    2p can be added or subtracted as many times as desired.
  2. Find an angle coterminal with 210o.

    3. 210o + 360o = 570o or 210o - 360o = - 150o
    Again, 360o can be added or subtracted as many times as desired.

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