Complex number to polar and cartesian form

Question

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?

Answer 1

Note that

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

thus

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

Answer 2

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) $\pi$.

Answer 3

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

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