Trigonometry problem

Cos (pi/7) also came up in this thread: Thread 65139. I got the result to the problem set in that thread (pi/7) by a numerical method, but was left wondering if there was an analytic one as pi/7 is a neat result where 0.4487 isn't. Can anyone see a link between the stuff above and that problem?

Doesn't look like it. If there were a simple expression for cos(pi/7), it would surely be listed here:

http://mathworld.wolfram.com/TrigonometryAnglesPi7.html

As that page notes, there's definitely no way to do it without using at least cube roots (7 isn't a Fermat prime, so cos(pi/7) isn't constructible). You'll have to settle for what comes out of the cubic formula.