We offer a slick, quick, elegant geometric solution as to why f(x) = cos(x). Our colleague Kurt Bryan, Department of Mathematics, Rose-Hulman Institute of Technology, saw this solution in a flash!

Note from the helpful figure at the start of the solution and below that OQ has length 1 and this means that OR has length cos(a). But OR and RT are the same length as the line QM is at 45 degrees with the x-axis. This makes RW also cos(a). And so if we roll out PQ - length a (arclength a as we are in a unit circle and angle QOP is a radians) until it is exactly PV (distance a) then the length of VN is the same as the length of RT, i.e. cos(a).