# Expressing polar complex numbers in cartesian form

I need to express $z = 4e^{-i\pi/3}$ in the form of $x + yi$ and represent it on the Argand diagram.

I think that $4 = \sqrt{x^{2} + y^{2}}$ and that $\tan (\pi/3) = y/x$ but I haven't been able to do anything useful with this information...

Is this solvable via simultaneous equations?

Thanks!

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In view of the answers posted, you should realize that it is MUCH easier computationally to convert from polar to rectangular form than to go the other way: $re^{i\theta} = (r\cos\theta) + i(r\sin\theta)$. You can do it in your head. – MPW Mar 4 '14 at 18:19

Using Euler Formula, $$e^{-\dfrac{i\pi}3}=\cos\left(-\frac\pi3\right)+\sin\left(-\frac\pi3\right)=\cos\left(\frac\pi3\right)-i\sin\left(\frac\pi3\right)=\frac12-\frac{\sqrt3}2i$$

Reference: the definition of $\displaystyle\arctan\frac yx$

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In my googling before posting this question, I did come across this method but avoided it since I haven't used it before. Seeing you doing it shows it's rather simple but I have a question - What happened to the 4 in front of the e? – Jacobadtr Mar 4 '14 at 17:30
@Jacobadtr, just multiply the modulus$(4)$ – lab bhattacharjee Mar 4 '14 at 17:31

Hint

Use the Euler's formula $$e^{i\theta}=\cos\theta+i\sin\theta$$

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