Unit Circle and Reference Angles

					                                   Unit Circle and Reference Angles
                               1 3                                               1 3
                              − ,  
                               2 2 
                                                             (0,1)                  ,  
                                                                                   2 2 
                                                                                     
                                               2π           π                 π
           2 2                                             2                                            2 2
         −                                    3                             3                                 
          2 , 2                                                                                         2 , 2             cosine
                 3π                               120○    90   ○      60○                 π                               value
                    4                                                                       4
                                  135○                                               45○
                                             Numerator is
   3 1                                                                                                          3 1
 −                                          always one                                                               
  2 , 2                                                             Numerator is                          π     2 , 2
        5π                                   less than                                                               
                                             denominator               always π                 30   ○
                                                                                                            6
           6       150○                         times π                                                                      sine value
                                                                                                  0        0
   (− 1,0)   π     180    ○

                                              Numerator is            Numerator is              360○       2π        (1,0)
             7π
                              ○                always one            always one less
                     210
  3 1                                        more than               than twice                         11π
−           6                                                                             330○                 3 1
 2 ,− 2                                     denominator             denominator                          6            
                                 225   ○
                                                 times π                 times π 315○                            2 ,− 2 
                                                                                                                        
                     5π
                                                240○                    300○
                      4                                                                    7π
       2     2                                            270○
     −   ,−                                                                               4             2     2
      2     2                                                                                             ,−   
                                            4π            3π                5π                          2    2 
                                                                                                                 
                                               3             2                 3
                           1     3                                                1    3
                           − ,−                                                    ,−   
                           2
                                2 
                                   
                                                             (0,−1)                 2
                                                                                        2 
                                                                                           


                                                            Value of Denominator                            Reference Angle
                                                                      ?
    Some hints when                                                   6                                          30○
  dealing with radians.                                               ?
                                                                      4                                          45○
                                                                      ?
                                                                      3                                          60○
A. Reference Angles:
                     π 
Example:    Draw 60°   in Quadrant II.
                     3
1. (a) We are in the Quadrant II with a reference angle of 60 degrees. The angle in radians has
a denominator of 3. Thus, the angle in Quadrant II is 120 degrees or 2π/3.
  (b)Formulas for Reference Angles:
      Note: Let θ be the reference angle given.



                   Quadrant II                 Quadrant I
                    180 - θ                       θ




                   Quadrant II                 Quadrant II
                     θ-180                       360- θ




        In this example, we in are in Quadrant II. Thus, the angle is 180 degrees-60 degrees =
        120 degrees.

Solution:

                    120º
                   (2π/3)



                          60º




Note: Knowing reference angles speeds up things considerably.
                            Positive and Negative Quadrants



                           S                                            A

                 Quadrant II                                      Quadrant I
                     (+)                                      All trigonometric
             sine and cosecant                                 functions are
                                                                    positive
                      (-)
               cosine secant
             tangent cotangent


                           T                                            C

                    Quadrant III                                    Quadrant IV
                         (+)                                             (+)
               tangent and cotangent                             cosine and secant

                           (-)                                            (-)
                    sine cosecant                                  sine cosecant
                    cosine secant                                tangent cotangent




All trigonometric functions are positive in Quadrant I


Sine and cosecant are positive in Quadrant II

Tangent and cotangent are positive in Quadrant III

Cosine and secant are positive in Quadrant IV
*Note: This information is used in conjunction with reference angles.
B. Using Reference Angles To Solve Trigonometric Equations.
                                  2
Example: Solve :    sin 2θ = −      .
                                 2
To solve this equation, take the inverse sine (arcsin) of both sides.
                                                                    2
                                       sin -1 (sin 2θ ) = sin −1  −
                                                                  2 
                                                                      
                                                                     
Note: 2θ means you have to make two revolutions around the unit circle. nθ determines
the number of revolutions.
Steps:
1. Determine which quadrants your desired angles lie in. In this example, sine is negative in
Quadrants III and IV.



                                   S                          A




                                   T                          C




2. Find corresponding angles in those quadrants. In this example, we need an angle in Quadrant
III and an angle in Quadrant IV which has a sine of
                2
           −      . The reference angle is 45 degrees which means the angles have a denominator
               2
of four.
           First time around:
           --Quadrant III (angle in radians is one more than denominator)
                   5π
                    4
           --Quadrant IV (angle in radians is 1 less and twice denominator)
                   7π
                    4
           Second time around (just add 2π to the previous angles):
           --Quadrant III (angle in radians is one more than denominator)
                   5π        13π
                      + 2π =
                    4         4
           --Quadrant IV (angle in radians is 1 less and twice denominator)
                    7π        15π
                       + 2π =
                     4          4
                           5π 7π 13π 15π
           Solution:          ,   , ,
                            4 4 4     4

				
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posted:10/21/2011
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