Sullivan Algebra and Trigonometry: Section 7.1 Angles and Their

							     Sullivan Algebra and
   Trigonometry: Section 7.1
   Angles and Their Measures
                  Objectives of this Section
• Convert Between Degrees, Minutes, Seconds, and Decimal
Forms for Angles
• Convert From Degrees to Radians, Radians to Degrees
• Find the Arc Length of a Circle
•Find the Area of a Sector of a Circle
A ray, or half-line, is that portion of a
line that starts at a point V on the line
and extends indefinitely in one direction.
The starting point V of a ray is called its
vertex.



             V          Ray
If two lines are drawn with a common
vertex, they form an angle. One of the
rays of an angle is called the initial side
and the other the terminal side.



               
   Vertex             Initial Side

      Counterclockwise rotation
      Positive Angle


    Vertex                 Initial Side
    Clockwise rotation       Negative Angle




Vertex                     Initial Side

    Counterclockwise rotation Positive Angle
An angle  is said to be in standard position
if its vertex is at the origin of a rectangular
coordinate system and its initial side
coincides with with positive x - axis.
                  y
   Terminal
   side
                       
              Vertex    Initial side     x
When an angle  is in standard position, the
terminal side either will lie in a quadrant, in
which case we say  lies in that quadrant,
or it will lie on the x-axis or the y-axis, in
which case we say is a quadrantal angle.

          y                         y



                           x
                                                      x


  is a quadrantal angle        lies in Quadrant III
Angles are commonly measured in either
Degrees or Radians
The angle formed by rotating the initial side
exactly once in the counterclockwise direction
until it coincides with itself (1 revolution) is
                                                 
said to measure 360 degrees, abbreviated 360 .
                         Terminal side
                         Initial side
    Vertex

                        1
     One degree, 1 , is     revolution.
                        360
                                     1 
A right angle is an angle of 90 , or
                                     4
revolution.

  Terminal
  side




     Vertex             Initial side
                      1
             90 angle;   revolution
                       4
 A straight angle is an angle of 180 ,
   1
 or revolution.
   2


Terminal side       Vertex   Initial side

                   1
          180 angle; revolution
                    2
           
 Draw   -135 angle.
Draw aa-135 angle.
                y



          Vertex      Initial side
                                     x
                       135
One minute, denoted, 1 , is defined as
1
   degree.
60
One second, denoted 1”, is defined as
 1                1
      minute, or      degree.
 60              3600


                                          
1 counterclockwise revolution = 360
        60 = 1
                       60 = 1
Convert 30 1255 to a decimal in degrees.
           


                                             
               30  12  1  55  1 
30 12 55  
   
                                      
                        60       3600

                30  0.2  0.015278
                                         



                30.215278
Convert 45.413 to D M S form.
                            


                    60
0.413  0.413    24.78
                
                    1
                60
0.78  0.78        46.8  47
                 1
     45.413  45 24 47 
                        
Consider a circle of radius r. Construct an
angle whose vertex is at the center of this
circle, called the central angle, and whose rays
subtend an arc on the circle whose length is r.
The measure of such an angle is 1 radian.



                                 r
                    
                          r
              1 radian
For a circle of radius r, a central angle of
radians subtends an arc whose length s is
                  s  r
Find the length of the arc of a circle of radius
4 meters subtended by a central angle of 2
radians.
     r  4 meters and  = 2 radians
     s  r  42  8 meters
The area A of the sector of a circle of radius r formed by a
central angle measured in radians is
                                              1 2
                                            A r 
                                              2

Example: Find the area of the sector of the circle of
radius 4 inches formed by an angle of 45 degrees.

              45
                                
              45                  rad
                       180       4

               1 2  
            A  (4 ) 
               2    4
            A  2  6.28            in 2
1 revolution = 2 radians


     180   radians
        


               
  1 degree =         radian
               180
               180
  1 radian =         degrees
               
             
Convert 135 to radians.
                      radian
      135  135 
                
                     
                  180
            3
            radian
            4
          2
Convert -     radians to degrees.
           3
     2               2 180 
        radians =        
      3                3 
                  120  