We never learned the unit circle like you did, but as far as I pieced together from various things I think I understand what you need to know, tell me if it's too much or too little.

Cobraroll explained the actual circle well, but I have a suspicion you're referring to the values on the unit circle rather than the concept itself.

Firstly radian measure. I won't go into the details of what it is because you may already know. The key thing to remember is pi is equal to 180 degrees. So 2pi = 360 degrees ; pi/2 = 90 ; pi/3 = 60 ; pi/6 = 30 and pi/4 = 45. Thus the formula really is if you have something in degrees, multiply by pi/180. If you have something in radians, multiply by 180/pi. An easy trick to remembering which is which is to look at the denominator, if what you have is in radians, multiply by the one with the pi on the bottom, vice versa for degrees

Now that that's settled, let's look at the unit circle.

Let P be a point on the unit circle of co-ordinates (x,y). Let P have an angle of Î¸ with the x-axis. Therefore, by pythagoras, given the hypotenuse is the radius of the circle and equal to one

cosÎ¸ = x/1 = x

sinÎ¸ = y/1 = y

tanÎ¸ = y/x

Thus the co-ordinates for P can be written as (cosÎ¸, sinÎ¸).

Now the horrible part... how to memorise the values of the co-ordinates when Î¸ is equal to various numbers.

Personally, since I wasn't taught this way, I'll teach you my way and how to convert it to the unit circle.

If Î¸ is equal to pi/4, then the other angle has to be pi/4 because the angles in a triangle must add up to pi (180). Thus the triangle is isoceles because the two base angles are equal. Let one of the smaller sides equal one, the other side must also equal one since opposite sides of an isoceles triangle are equal. Now what about the hypotenuse? well by pythagoras the hypotenuse = square root of 1^2 + 1^2. The hypotenuse = square root of 2.

But the hypotenuse of the unit circle isn't the square root of 2, it's 1. Therefore we have to make the hypotenuse of this triangle 1. Look at the bottom left triangle. All the sides have been divided by the square root of 2. This is fine since I just let the two sides equal one, the ratio and pythagoras and everything still holds if I divide it by root 2. Except now we have the two co-ordinates that define the point for the unit circle.

Now we can use trigonometry! We found earlier that cosÎ¸ = x, sinÎ¸ = y, tanÎ¸ = y/x. If we substitute the values we have on the triangle

sin(pi/4) = 1/root 2 ; cos(pi/4) = 1/root 2 ; tan(pi/4) = 1/root 2 divided by 1/root 2 = 1

The other two values, pi/3 and pi/6 I have shown on the triangles below. For pi/6, you can just swap the values, as long as you know which side of the triangle represents x and which represents y you'll be fine.

Now, this is only for the first quadrant. For the other three you'll be pleased to know it's just these exact same triangles mirrored. For example, in the second quadrant, the x value will be negative, but the y value will be positive. So cosÎ¸ is negative, sinÎ¸ is positive, and tanÎ¸ is negative.

The trick to remembering the angles, I find, is to remember it's the same angle, but taken from the x axis. So for the third quadrant,

In that sense if we have an angle of Î¸ in the first quadrant, the equivalent angles are:

Second Quadrant: pi - Î¸

Third Quadrant: pi + Î¸

Fourth Quadrant: 2pi - Î¸

Hope that helps.