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Given an complex number $z=a+bi$, we could find the polar form. For example $z=1+\sqrt{3}i$ has radius $2$ and $\arg z=\frac{\pi}{3}$.
My question is: Do I have to memorize all the possible sin and cosine values for every possible argument $\theta$? How can I know without a calculator that $\cos \theta$ = 1/2 and $\sin\theta=\frac{\sqrt{3}}{2}$ satisfies $ \theta = \frac{\pi}{3}$ ? How can I memorize this values efficiently?
There are an infinity of values... But you mean the common values $\pi$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$. It is easy to memorize them as follows: $$ \begin{array}{c|c|c} \theta& \sin\theta & \cos\theta \\ \ \\ \hline 0 & \frac{\sqrt0}2&\frac{\sqrt4}2\\ \hline\frac\pi6 & \frac{\sqrt1}2&\frac{\sqrt3}2\\ \hline\frac\pi4 & \frac{\sqrt2}2&\frac{\sqrt2}2\\ \hline\frac\pi3 & \frac{\sqrt3}2&\frac{\sqrt1}2\\ \hline\frac\pi2 & \frac{\sqrt4}2&\frac{\sqrt0}2\\ \end{array} $$
