As one can see here, in this unit circle (like this one: http://www.coolmath.com/sites/cmat/files/images/28-trigonometry-03.gif), 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.
Let us start from figure 1 with all the given as shown. we need the answer the following questions (preferably in sequence):-
1) $\angle C = ?$; 2) $BC = ?$; and 3) $AC = ?$.
Figure 2 shows an equilateral triangle with sides = 2. The green triangle is just half of it. Then,
1) $\angle Q = ?$; 2) $\angle QPR = ?$; $QR = ?$ and 4) $PR = ?$.