SOLUTION: How do I prove cos(pi/6)=sqrt(3)/2 knowing that sin(pi/6)=1/2?

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Question 786238: How do I prove cos(pi/6)=sqrt(3)/2 knowing that sin(pi/6)=1/2?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If you were taught that
%28sin%28x%29%29%5E2%2B%28cos%28x%29%29%5E2=1 you would use that and the fact that pi%2F6 is in the first quadrant, where sine and cosine are positive.

Otherwise, if you were only taught that sine and cosine are trigonmetric ratios that apply to right triangles, use a right triangle with a pi%2F6 angle and hypotenuse length 1. Then invoke the Pythagorean theorem. The measures of the legs of that right triangle are cos%28pi%2F6%29 and sin%28pi%2F6%29.

In fact, if you split an equilateral triangle in half using a median (connecting the midpoint of one side to the opposite vertex), you would get two congruent right triangles with one pi%2F6 angle. Fron that idea, you can deduce both values, sin%28pi%2F6%29 and sin%28pi%2F6%29. The shorter leg of those traingles, opposite the pi%2F6 angle is half of the side of the equilateral triangle, that is now the right triangle's hypotenuse. Hence sin%28pi%2F6%29=1%2F2