Find the modulus of each of the following complex numbers and hence express each of them in polar form: –i
Let $Z=-i=r(\cos \theta+i \sin \theta)$
Now, separating real and complex part, we get
0 = rcosθ……….eq.1
-1 = rsinθ …………eq.2
Squaring and adding eq.1 and eq.2, we get
$1=r^{2}$
Since r is always a positive no., therefore,
r = 1,
Hence its modulus is 1.
Now, dividing eq.2 by eq.1, we get,
$\frac{r \sin \theta}{r \cos \theta}=\frac{-1}{0}$
$\operatorname{Tan} \theta=-\infty$
Since $\cos \theta=0, \sin \theta=-1$ and $\tan \theta=-\infty$. Therefore the $\theta$ lies in fourth quadrant.
$\operatorname{Tan} \theta=-\infty$, therefore $\theta=-\frac{\pi}{2}$
Representing the complex no. in its polar form will be
$\mathrm{Z}=1\left\{\cos \left(-\frac{\pi}{2}\right)+\operatorname{isin}\left(-\frac{\pi}{2}\right)\right\}$