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

Question:

Find the modulus of each of the following complex numbers and hence express each of them in polar form: –i

 

Solution:

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

$\mathrm{Z}=1\left\{\cos \left(-\frac{\pi}{2}\right)+\operatorname{isin}\left(-\frac{\pi}{2}\right)\right\}$

 

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