Question

If $z=\sin \theta +i(\cos \theta -1),i^2=-1$ is a purely real as well as a purely imaginary number, then the number of value(s) of $\theta \in [0,4\pi ]$ is

Answer 1

Given : $z=\sin \theta +i(\cos \theta -1)$
This is a purely real as well as a purely imaginary number, so
$Re\left(z\right)=0,Im\left(z\right)=0$
$\Rightarrow \sin \theta =0\Rightarrow \theta =0,\pi ,2\pi ,3\pi ,4\pi$
And
$\Rightarrow \cos \theta -1=0\Rightarrow \cos \theta =1\Rightarrow \theta =0,2\pi ,4\pi$
From $\left(1\right)$ and $\left(2\right)$, we get
$\theta =0,2\pi ,4\pi$

Hence, the number of values of $\theta \in [0,4\pi ]$ is $3.$