Question

Express the following complex numbers in the form $r\left(\cos \theta +i\sin \theta \right)$ :
tan$\alpha$ - i

Answer 1

Let $tan\alpha =k$......[$k$ is a constant]

So, $z=k-i$

Now, $r=|z|=\sqrt{k^2+1}$ and $\theta =ta{n}^{-1}\frac{-1}{k}$

Hence, $z=\sqrt{k^2+1}\left[cos\right(ta{n}^{-1}\frac{-1}{k})+isin(tan--1\frac{-1}{k}\left)\right]$

$z=\sqrt{tan{\alpha }^2+1}\left[cos\right(ta{n}^{-1}\frac{-1}{tan\alpha })+isin(tan--1\frac{-1}{tan\alpha }\left)\right]$