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the cos(60) is 1/2 the sin(60) is sqrt(3)/2.
You prove it by drawing an equilateral triangle; that is a triangle that has all 3 sides equal. To make it easy on ourselves, lets make the sides equal to 2.
A fact about triangles is that all of their interior angles have to add up to 180 degrees. Since all the sides are equal, all the angles of our triangle are equal as well.
60 + 60 + 60 = 180.
Now take a drop a line straight down from the top point and connect it to the middle of the base. We have just made 2, 90 degree triangles. Each of the have a 60 degree angle and a 90 degree angle which leaves us with the last angle being 30 degrees (which is half of the 60 degree angle that we cut thru).
So what are our measurements now? we have an outside (the hypotenuse) equal to 2; and we have a leg equal to 1.
Use the pythagorean theorum to figure out the missing leg.
a^2 + b^2 = c^2
a^2 + 1^2 = 2^2
a^2 = 4-1 =3
sqrt(a^2) = sqrt(3)
the other leg has to be equal to the squareroot of 3 (sqrt(3)).
Now we have to know the definitions for cos and sin. Cosine (cos) means the adjacent leg over the hypotenuse. The adjacent leg to our 60 degree angle is 1, and the hypotenuse is 2.
So the cos(60) = 1/2.
Sine(sin) means the opposite leg over the hypotenuse. The opposite leg from the 60 degree angle is sqrt(3) and the hypotenuse is still 2.
sin(60) = sqrt(3)/2
Does this help?
amistre64 it sure helped me, I was trying to figure out how to help, you made it quite clear.