Download Reference Angles

Reference Angles
A reference angle for an angle θ in standard position is the positive acute angle
between the terminal side of θ and the (closest part of the) x-axis. In these
notes, the reference angle for θ will be denoted  .
EXAMPLE 1: A reference angle for   27 is    27 , because   27 is
already an acute angle.
EXAMPLE 2: A reference angle for   125 is    180  125  55 because
  125 is in the second quadrant, and the closest part of the x-axis is the
negative x-axis, or 180 , which is larger than   125 .
EXAMPLE 3: A reference angle for   190 is    190  180  10 , because
  190 is in the third quadrant. Again, the negative part of the x-axis is closest
to   190 , but this time   190 is larger than 180 .
EXAMPLE 4: A reference angle for   322 is    360  322  38 . This is
because   322 is in the fourth quadrant, and the positive x-axis is the closest
to θ. (The positive x-axis is the terminal side of both 0 and 360 .)
The figure below summarizes the method for finding a reference angle for θ if
0    360 (0    2 )
We study reference angles because of the following theorem:
Reference Angle Theorem
If   is the reference angle for  , then the trigonometric functions of   are
the same as those for  except for a possible change in  sign.
(That is, sin     sin  , cos     cos  , tan    tan and so on.)
1
. The angles between 0 and 360 that
2
have a reference angle of 60 are 60 in Quadrant I, 120 in Quadrant II, 240
in Quadrant III, and 300 in Quadrant IV. As you can see from the figure below,
x 1
x
1
cos 60  cos 300   , and cos 120  cos 240    . The values differ
r 2
r
2
For example, we know that cos 60 
only in sign.
EXAMPLE 9: Find tan 210 .
Solution: First, we find a reference angle for   210 and determine the sign of
tan 210 . Since   210 is in the third quadrant,    210  180  30 , and
tan 210  0 . We know that tan 30 
1
3

3
3
and, therefore, tan 210 
3
3
also.
EXAMPLE 10: Find sec 135 .
Solution: First, we note that   135 is in the second quadrant. Thus,
1
 0 , and    180  135  45 . We find
cos 135
1
2
sec   sec 45 

 2 and conclude that sec 135   2 .

cos 45
2
sec 135 
EXAMPLE 11: Find a fourth quadrant angle in degrees such that sin   
Solution: First, we find the reference angle,   . The acute angle such that
2
is    45 , so the fourth quadrant angle we are looking for is
2
  360  45  315 .
sin   
2
.
2
EXAMPLE 12: Find a second quadrant angle in radians such that cos   
1
.
2
1
would be    60  . The
2
angle we are looking for, then is   180  60  120 .
Solution: The acute angle,   , such that cos   
You can also find reference angles for negative angles and angles over one
revolution:
EXAMPLE 13: Find a reference angle for   115 .
Solution:   115 is in the third quadrant. Its reference angle
is ˆ  180  115  65 .
EXAMPLE 14: Find a reference angle for   1790 .
Solution: 1790  4(360)  350 , so   1790 is coterminal with 350 and
therefore has the same reference angle. A reference angle for 350 is
   360  350  10 .