<s>[INST] Remember trig identities like $\cos(\pi/3) = 1/2$ I have started doing complex analysis and I keep having to switch between rectangular co-ordinates and polar form and I keep running into stuff like -  $\cos(\pi/3) = 1/2$.
I keep having to look these up. Am I expected to memorize these or what, its getting to be a serious pain having to keep looking them up. Do people generally just memorize these identities for complex analysis?
 [/INST] All you really need to know are the values of $\sin(x)$, where $x$ is $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$, and $\frac{\pi}{2}$. All of the other values come directly from these:
$$
\begin{align}
 \sin(0) &= \frac{\sqrt{0}}{2} = 0
\\
\\
\sin(\pi/6) &= \frac{\sqrt{1}}{2} = \frac{1}{2}
\\
\\
\sin(\pi/4) &= \frac{\sqrt{2}}{2}
\\
\\
\sin(\pi/3) &= \frac{\sqrt{3}}{2}
\\
\\
\sin(\pi/2) &= \frac{\sqrt{4}}{2} = 1
\end{align}
$$
These value are easy to memorize, or you can just find them using Gerry's or Lopsy's answers.
Now, say you want to find $\cos(\pi/3)$.  Since $\sin(\pi/3) = \frac{\sqrt{3}}{2}$, and $\sin^2(x) + \cos^2(x) = 1$, you get that
$$
\cos^2(\pi/3) = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4}.
$$
Hence, $\cos(\pi/3) = \pm \frac{1}{2}$.  Since $\pi/3$ is in the first quadrant, it must be the positive value.
This might be the long way around (and perhaps looking at triangles or the unit circle is easier), but this saves you from looking up any values.
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