# Complex number to polar and cartesian form

I need to tranform: $$z:=-4e^{{\pi}i/3}$$ to the polar (I know it's almost polar) and cartesian form, i.e. find x and y coordiantes.
$$-4e^{{\pi}i/3}=-4(\cos(\frac\pi3)+\sin(\frac\pi3)i)=4(-1)(\cos(\frac\pi3)+\sin(\frac\pi3)i)=4(-\cos(\frac\pi3)-\sin(\frac\pi3)i)$$ Don't really know how to continue.
So what's the trick behind this complex number?

• Just put $\cos(\pi/3) = 1/2$ and $\sin(\pi/3) = \sqrt{3}/2$. Commented Feb 7, 2018 at 15:35
• There's little to continue. $z$ started out in polar form. To finish cartesian, just distribute the $4$ over the parentheses in your last expression to find the real and imaginary parts. You might explicitly evaluate the trig functions in terms of $\sqrt{3}$. Commented Feb 7, 2018 at 15:36
• it is $$z=-2+2i\sqrt{3}$$ Commented Feb 7, 2018 at 15:40
• Usually by definision the form $$z:=\rho e^{i\theta}$$ is defined with $\rho \ge0$
– user
Commented Feb 7, 2018 at 15:41
• @GNUSupporter will remember to do it in the future Commented Feb 7, 2018 at 15:44

Note that

• $\cos(\frac\pi3)=\frac12$
• $\sin(\frac\pi3)=\frac{\sqrt3}2$

thus

$$4\left(-\cos(\frac\pi3)-\sin(\frac\pi3)i\right)=-2-2\sqrt3\,i$$

Multiplying a complex number by $-1$ is the same as rotating it (if we think of it as an arrow from the origin) 180 degrees in the complex plane, to make it point in the opposite direction. So you can remove the minus sign and compensate the angle by adding (or subtracting) $\pi$.

Continue your work by substituting $\sin \frac\pi3 = \frac{\sqrt3}{2}$ and $\cos\frac\pi3 = \frac12$.

\begin{aligned} -4e^{{\pi}i/3}&=-4(\cos(\frac\pi3)+\sin(\frac\pi3)i)\\ &=4(-1)(\cos(\frac\pi3)+\sin(\frac\pi3)i)\\ &=4(-\cos(\frac\pi3)-\sin(\frac\pi3)i)\\ &=4\left(-\frac12-\frac{\sqrt3}{2}i\right)\\ &=-2-2\sqrt3 \,i \end{aligned}