Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$.
I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\,$ is undefined, can I just say $\theta$ is $\dfrac{\pi}{2}$? So $z$ in polar form will be ${4e}^{i\pi/2}$.
Thank you for your help!
$\endgroup$ 41 Answer
$\begingroup$Let $Z=-4i$
That is $Z=4[0+(-1)i]$
You have to find a theta value such that $\sin \theta =-1$ and $\cos \theta =0 $
Since $\cos \frac{3\pi}{2}=0$ and $\sin \frac{3\pi}{2}=-1 $
Thus $$Z=4 \left[\cos \frac{3\pi}{2}+i \sin \frac{3\pi}{2} \right]$$
Since $$e^{i\theta}=\cos \theta +i \sin \theta$$
$$Z=4 e^{i\frac{3\pi}{2}}$$
$\endgroup$ 0
