Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2i
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
$Z=2\left\{\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right\}$