Background Knowledge – The Unit Circle, Trigonometric Ratios and Angles in Radians
I. Families of Angles on the Unit Circle




A. 0o = 0 radians; 30 =
rads ; 45 =
rads ; 60 =
rads ; 90 =
rads
6
4
3
2
B. The memorized sines and cosines of all angles in these families come from the
two right triangles below.
1
2
2
1
2
1
30o
45o
3
2
2
2
C. By locating angles in standard position, the opposite side extends in the “y”
direction and the adjacent in the “x” direction, meaning the coordinates of a
point on the unit circle are cos  , sin   .
D. If we can remember that from smallest to largest, the possible “side lengths”
2
3
1
of the “triangles” in the 5 families are 0, ,
,
, 1, we can then use the
2 2
2
unit circle to remember many facts about trigonometric ratios.
(0,1)
sin 
 1 3
 ,

 2 2 


1 3
 ,

2 2 



2 2


 2 , 2 



3 1


 2 ,2


3
2
(-1,0)
2
2
1
2
2
2

3 1


 2 , 2 


 2 2


 2 , 2 


 3 1


 2 ,2


cos (1, 0)
 3 1


 2 , 2 


 2
2


 2 , 2 



2
2


 2 , 2 


 1
3
  ,

 2

2


1
3
 ,

2
2 

(0,-1)
II. The Other 4 Trigonometric Functions
A. If the unit circle and an understanding of the families of angles let us evaluate
2
 5  
 , evaluating tangent,
trigonometric ratios like cos 
  the answer is 2 
 4  
cotangent, secant and cosecant require us to practice and memorize a few
calculations involving the numbers from the unit circle.
1. tan  
opposite sin 

adjacent cos 
2. sec  
hypotenuse
1

adjacent
cos 
3. cot  
adjacent cos 

opposite sin 
4. csc  
hypotenuse
1

opposite
sin 
All of the calculations you’ll need are listed below. The best idea is to aspire to
eventually be able to know the answer once you see the original fraction. To get to that
point, you might need to do the calculations a few times for yourself.
1
 undefined
0
1
2
3
2
3
=
3
1
1
2
=2
2
2
2
2
=1
1
2
2
=
3
2
1
2
1
2
3
2
=
3
=
2 3
3
1
=1
1
0
0
1
III. Positive and Negative Angles and Converting to Radians or Degrees
A. Positive angles start at the positive x axis and rotate counterclockwise.
B. Negative angles start at the positive x axis and rotate clockwise.
C. When you convert angles, the result should match in size and direction.
1. You should use families of angles whenever possible, but might have
to rely on a conversion factor if the angle doesn’t have a memorized
reference angle.
Ex: Convert 765o to radians.
Think: 765 – 720 = 45. So it is 2 full rotations plus a
4 +

17
=
4
4

4

radians have a lot in common, because they are co4
terminal. They have the same trig. functions, the same reference

angle and are in the same quadrant. However, 765o 
radians.
4
765o and
11
to degrees.
15
Think: 180o and  radians are equivalent, so multiplying by
180

or
just changes units of measure.

180
Ex: Convert 
12o
11 radians
180
11 radians
180



=
= -132o
15
 radians
15
 radians