Understanding and Memorizing the Unit Circle

The unit circle is one of the most important concepts to understand in Trigonometry.
As a tutor who emphasizes understanding and comprehension over memorization, I try to make it as easy as possible for my students.
Here's the way I like to look at it:
1) First, realize that the unit circle is simply a few points drawn on an graph with an x-axis and a y-axis.
2) Recognize that there is an overall pattern.
  • Every 90 degrees (0, 90, 180, 270) is a combination of 0 and 1 (positive and negative).
  • Every 45 degrees (45, 135, 225, 315) is √2/2 (positive and negative).
  • Every 30 degrees (30, 60, 120, 150, 210, 240, 300, 330) are combinations of 1/2 and √3/2 (positive and negative).
This means that you only have to remember three numbers: 1/2, √2/2, and √3/2 (positive and negative).
The first quadrant (0-90 degrees), has all positive numbers, just like you'd expect in any other graph.
The second quadrant (90-180 degrees), has positive y-values (sin values) and negative x-values (cos values), just like you'd expect in any other graph.
The third quadrant (180-270 degrees), has all negative numbers, just like you'd expect in any other graph.
The fourth quadrant (270-360 degrees), has negative y-values (sin values) and positive x-values (cos values), just like you'd expect in any other graph.
So the hardest part is to remember when your 30 degree angles will be 1/2 or √3/2. The way I like to think about it is that we know that √3 is greater than 1. So when you look at a graph of the unit circle, the 30 degree angles will either be wide and short or tall and skinny. The longer side will always be √3/2 and the shorter side will always be 1/2.
You can use a simpler, easier to remember shortcut that I figured out recently which goes like this:
0 degrees: √0/2 (which = 0)
30 degrees: √1/2 (which = 1/2)
45 degrees: √2/2
60 degrees: √3/2
90 degrees: √4/2 (which = 1)
As you can see, as you increase your angle, the pattern is that the radicand increases by 1 for each point on the unit circle.
I hope this helps!


Once you know the coordinates of the intersection of the coordinate axes with the unit circle and the coordinates of the 30°, 45° and 60° points in the first quadrant, the coordinates of nine other points are easily found by using the symmetry of the unit circle.  It helps very much to make a sketch of the unit circle.  For example the first coordinate of the 120° point is the opposite of the first coordinate of the 60° point and the second coordinate of the 120° point is the same as the second coordinate of the 60° point.  This means that cos(120°) (= cos(2/3 Pi) = -1/2 and sin(120°)=√(3)/2.  In general, if you rotate the point (1,0) on the unit circle about the origin with a magnitude of "m" the coordinates of the image are by definition (cos(m), sin(m)).  This is a way to get values of the circular functions (traditionally called the trigonometric functions) without the use of right triangles. 
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