Find the modulus of each of the following complex numbers and hence

Solution:

Let $Z=2 i=r(\cos \theta+i \sin \theta)$

Now, separating real and complex part, we get

0 = rcosθ ……….eq.1

2 = rsinθ …………eq.2

Squaring and adding eq.1 and eq.2, we get

$4=r^{2}$

Since r is always a positive no., therefore,

$r=2$

Hence its modulus is 2.

Now, dividing eq.2 by eq.1, we get,

$\frac{r \sin \theta}{r \cos \theta}=\frac{2}{0}$

$\operatorname{Tan} \theta=\infty$

Since $\cos \theta=0, \sin \theta=1$ and $\tan \theta=\infty$. Therefore the $\theta$ lies in first quadrant.

$\tan \theta=\infty$, therefore $\theta=\frac{\pi}{2}$

Representing the complex no. in its polar form will be

$Z=2\left\{\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right\}$