Thinking Mathematically by Robert Blitzer Angles and Their Measure by liaoqinmei


									Angles and Their
   Students will be able   to   describe angles.
   Students will be able   to   use radian measure.
   Students will be able   to   us degree measure.
   Students will be able   to   use angles to model and solve
    real-life problems.
An angle is formed by two rays that have a
common endpoint called the vertex. One ray is
called the initial side and the other the terminal
side. The arrow near the vertex shows the
direction and the amount of rotation from the initial
side to the terminal side.
                  C                           A

          Terminal Side                    Initial Side
    Angles of the Rectangular Coordinate System

An angle is in standard position if
• its vertex is at the origin of a rectangular coordinate system and
• its initial side lies along the positive x-axis.

                    y                                       y
                             is positive
  Side                                            Vertex
                                                                   Initial Side

                                       x                                          x
                                            Terminal              
                        Initial Side
                                                                       is negative

Positive angles rotate counterclockwise.       Negative angles rotate clockwise.
Coterminal Angles

An angle of xº is coterminal with angles of
xº + k · 360º
where k is an integer.
The initial and terminal sides are the same.
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.
Radian Measure
Consider an arc of length s on a circle or radius
The measure of the central angle that intercepts
the arc is
 = s/r radians.

The circumference is              r

2πr units                             O   r

See Pg. 135
     Finding Complements and
• Angles are complementary if their sum is
• Angles are supplementary if their sum is π.

• Pg. 142 #18a, 20a, 22a, 24, 30, 32a, 34a
    Measuring Angles Using Degrees
       The figures below show angles classified by
their degree measurement. An acute angle measures
less than 90º. A right angle, one quarter of a complete
rotation, measures 90º and can be identified by a small
square at the vertex. An obtuse angle measures more
than 90º but less than 180º. A straight angle has
measure 180º.
                      90º                             180º

   Acute angle    Right angle    Obtuse angle     Straight angle
   0º <  < 90º   1/4 rotation   90º <  < 180º    1/2 rotation
Conversion between Degrees
and Radians
•   Using the basic relationship
•    radians = 180º
•   2π radians = 360º
        = (one complete rotation)

•   To convert degrees to radians, multiply
    degrees by ( radians) / 180
•   To convert radians to degrees, multiply radians
    by 180 / ( radians)
Convert each angle in degrees to
            Example cont.
            40  40 

                      180 9
                     5    
        75  75     
                  180 12

         8  8 180            o

                       160o
          9    9  rads
         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                                  s

Pg. 144 # 90, 96, 98
                                       O   r
          Area of a Sector of a Circle
For a circle of radius r, the area A of a sector of the
  circle with central angle  (in radians) is given by
               1 2
             A r 


Pg. 144 #102
Angular Speed
 Consider a particle moving at a constant
   speed along a circular arc of radius r. If
   s is the length of the arc travelled in
   time t, then
                      arch length   s
 Linear Speed                   
                        time    t

 If Ѳ is the angle measure corresponding to
    arc length s then
 Angular Speed          central angle 
                                   
                         time       t
• Groups: Pg 143 – 145 # 18b, 22b,
  32b, 44, 56, 70, 76, 92, 104, 112

• Homework: 17 – 33 odd, 41 – 79 odd,
  89 – 103 odd, 111

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