As one can see here, in this unit circle (like this one: ), and in the table that I linked to on Wikipedia below, there are several points on the unit circle at which one can use a radical divided by a value (for sine and cosine, what I ask about, that value is 2) to obtain an exact value for the sine or cosine of an angle. An example is for $\frac{\pi}{3}$ radians, where is sine of that angle is $\frac{\sqrt 3}{2}$.

What I do not understand is where the idea to use the square root of 3 came from, and equally so, where the idea to divide the value by 2 came from. I do understand that the unit circle has a radius of 1 and sides of triangles made within it must pertain to the pythagorean theorem (hence these values with radicals, for accuracy), but that is all I understand.

How would one know to put exactly $\frac{\sqrt 3}{2}$ for the sine of $\frac{\pi}{3}$ radians? This is unclear to me.