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: $\sqrt{3}+\mathrm{i}$

 

Solution:

Let $Z=\sqrt{3}+i=r(\cos \theta+i \sin \theta)$

Now, separating real and complex part, we get

$\sqrt{3}=\operatorname{rcos} \theta \ldots \ldots \ldots . . \mathrm{eq} .1$

$1=r \sin \theta \ldots \ldots \ldots \ldots .$ 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,

$\frac{r \sin \theta}{r \cos \theta}=\frac{1}{\sqrt{3}}$

$\operatorname{Tan} \theta=\frac{1}{\sqrt{3}}$

Since $\cos \theta=\frac{\sqrt{3}}{2}, \sin \theta=\frac{1}{2}$ and $\tan \theta=\frac{1}{\sqrt{3}}$. Therefore the $\theta$ lies in first quadrant.

$\operatorname{Tan} \theta=\frac{1}{\sqrt{3}}$, therefore $\theta=\frac{\pi}{6}$

Representing the complex no. in its polar form will be

$\mathrm{Z}=2\left\{\cos \left(\frac{\pi}{6}\right)+\operatorname{isin}\left(\frac{\pi}{6}\right)_{\}}\right.$

 

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