# 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○
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.

180 - θ                       θ

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.

S                                            A

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

T                                            C

(+)                                             (+)
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

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:
5π
4
7π
4
Second time around (just add 2π to the previous angles):
5π        13π
+ 2π =
4         4
7π        15π
+ 2π =
4          4
5π 7π 13π 15π
Solution:          ,   , ,
4 4 4     4

```
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