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For example solve $\cos{\left(\frac{5\pi}{4}\right)}$ without a calculator or solve $\cos{(x)} = -\frac{1}{2}$.

I remember vaguely that the method involves referring to a triangle, but im not sure. Could you be kind enough to either explain how to go about these questions or to forward me to a website which explains this in detail.

Thanks a lot!

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    @John, both are valid solutions, unless you have extrinsic restrictions of the range of $x$. (You can also add or subtract any multiple of $2\pi$, of course).2011-09-05

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R. Knott's page might be of interest to you. You can go directly to the derivation using triangles.

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It's just you have to remember several well-known equalities. Such as:

$\begin{align*} \sin\frac{\pi}{4} &= \frac{\sqrt{2}}{2}\\ \sin\frac{\pi}{6} &= \frac{1}{2}\\ \sin\frac{\pi}{3} &= \frac{\sqrt{3}}{2} \end{align*}$

Plus - some basic facts, such as how sin/cos related, what happens if you add pi to the argument and etc.