How do you memorize the Unit Circle for trigonometry?

I think this will surely help you
It is a video on "A Trick to Remember Values on The Unit Circle"

If YOU GET ORGANIZED...

You only need the principal trig functions (sine, cosine, tangent) for
0, 30, 45, 60, and 90 degrees, which are respectively 0, ?/6, ?/4, ?/3, ?/2 radians.
[put numbers onto a 30-60 triangle; and on a 45-45 triangle]

Know the reciprocals of the principal functions (a principal function and its reciprocal make a "co-nonco" pair".  Know the sign nature of the functions in each quadrant (memory crutch:  ASTC).

The best way for me was to memorize the triangles for 0, 30, 45, 60, 90 degrees. The reason is that way you can "visualize" it in your mind, and I think it makes more sense that way. For example, on 30 degrees the base is longer than the height, and on 60 degrees the height is longer than the base. On 45 degree, height and base are equal.

I'm not so clear on what you mean about 2 pi to pi/6, and so on. For pi/6 which equals 30 degrees, I still picture the triangle, with the base being longer than the height. Remember that  pi/6 is a smaller angle than  pi/3 which is 60 degrees. Then pi/4 is between them at 45 degrees.

You don't really have to memorize everything on the unit circle. If you can memorize the 1st Quadrant, then you can apply those to the other Quadrants:

http://img.sparknotes.com/content/testprep/bookimgs/sat2/math2c/0003/30=pi6.gif

Just 3 angles in the first quadrant ( 0º to 90º, 0 to ?/2), their sine and the fact that 180º = ? radians is all you need to remember.

30º (1/6 of 180º, ?/6) sine 30º = 1/2

45º (1/4 of 180º, ?/4) sine 45º = ?2/2

60º (1/3 of 180º, ?/3) sine 60º = ?3/2

The cosine values you get from cos =?(1-sin^2).  For the above angles you get ?3/2, ?2/2, 1/2

Notice the pattern, the bigger the angle, the smaller the cosine, the larger the sine.

Notice that the sin and cos for 0º and 90º value can just be read off the circle.

Now comes the fun part: between 90º and 180º the sine doesn't change sign, so a 30º angle from 180º (i.e. 150º) has the same sine value as sine 30º, while the cosine changes sign.

Repeat this process around the circle keeping track of where you are on the unit circle so as to nail the signs correctly.

Tan, cot, sec, cosec are just gotten from the sine and cos values, you could memorize them but they are are not worth the effort.  Just know how to derive them and it will become second nature in  no time.

Draw a unit circle and mark the first quadrant and practice moving around it.

If YOU GET ORGANIZED...

You only need the principal trig functions (sine, cosine, tangent) for
0, 30, 45, 60, and 90 degrees, which are respectively 0, ?/6, ?/4, ?/3, ?/2 radians.
[put numbers onto a 30-60 triangle; and on a 45-45 triangle]

Know the reciprocals of the principal functions (a principal function and its reciprocal make a "co-nonco" pair". Know the sign nature of the functions in each quadrant (memory crutch: ASTC).