Question

How do I memorize the unit circle

Answer 1

All you have to do is memorize quadrant 1.  If you memorize quadrant 1, then you can figure out the rest of the unit circle by reasoning through it.  

So starting from the top, 90 degrees, and moving towards the right, down to 0 degrees, you will come across 3 angles, 60 degrees 45 degrees, and 30 degrees.

They follow a pattern:
(√1/2, √3/2)  (60 degrees)
(√2/2, √2/2) (45 degrees)
(√3/2, √1/2) (30 degrees)

Notice the 1,2,3 pattern and the 3,2,1 pattern.  And that each of them is a fraction with a 2 in the denominator.

If you can memorize the above, then the rest of the unit circle is just those values, but they make be positive or negative depending on which quadrant you are in.

If you understand how the x-y coordinates work, then you should be able to figure out the positive or negative aspect of the unit circle. 

For example, in quadrant 1, both x and y are positive.  In quadrant 2, the x is negative and the y is positive.  So if I wanted to know angle 150 degrees.  I recognize it is going to be the same point as the 30 degrees which is (√3/2, √1/2).  But since it is in quadrant 2, the x is negative.  Thus for 150 degrees, the unit circle says (-√3/2, √1/2)

Another example, 315 degrees will have the same point as 45 degrees which is (√2/2, √2/2).  But since it is in quadrant 4, the x is positive and the y is negative.  Thus, the unit circle for 315 degrees is (√2/2, -√2/2)

I hope that helps!