# Where do the radical expressions for the trig functions of various rational multiples of $\pi$ come from?

So if you look on the Wikipedia page for "Trigonometric constants expressed as real radicals", you will get a boatload of formulas for the trig functions of various rational multiples of $$\pi$$.

My question is, how were all these formulas deduced and/or derived? What is the proof that these formulas are correct? And are there any more potential formulas like this that can be derived, but not found by a mathematician?

• Can you link to the page or write a few of the formulas down? Usually a brief explanation is given as to how to the formulas are derived; regarding formulas that might exist undiscovered...probably, but if nobody has found it, we couldn’t know. – Clayton Dec 9 '18 at 14:47
• The values are typically derived by taking the simplest cases (for, say, $\pi/2$, $\pi/3$, $\pi/4$, and others that arise from various constructible regular polygons), hitting them with half-angle identities to get really small values, and then using multiple-angle identities. It's a pretty mechanical process, and more-or-less how trig tables were generated for centuries before we had calculators, so there's not much new to do in this space. However, finding "nice" forms and/or patterns, can be an interesting challenge. See this answer. – Blue Dec 9 '18 at 14:59
• You can see from the banner that the "page might contain original research". – John Glenn Dec 9 '18 at 15:09