If the hypotenuse is not 1, then the x and y sides will not be sine and cosine, but rather a scaled version of them.
Remember that sine and cosine represent ratios, so if the hypotenuse is r, then (x,y)=(rsin(a), rcos(a)), where 'a' is the angle. By pythagorian theorem, x^(2) + y^(2) = r^(2), note that 'r' can be divided on both sides to make the right side 1.
RootedPopcorn t1_j6eiydy wrote
Reply to comment by 321ECRAB123 in eli5: why dies the pythagorean identity work always if it is made with unit circle in mind? by [deleted]
If the hypotenuse is not 1, then the x and y sides will not be sine and cosine, but rather a scaled version of them.
Remember that sine and cosine represent ratios, so if the hypotenuse is r, then (x,y)=(rsin(a), rcos(a)), where 'a' is the angle. By pythagorian theorem, x^(2) + y^(2) = r^(2), note that 'r' can be divided on both sides to make the right side 1.