Find the modulus of each of the following complex numbers and hence express each of them in polar form: –1 + i
Let $Z=1-i=r(\cos \theta+i \sin \theta)$
Now, separating real and complex part, we get
-1 = rcosθ ……….eq.1
1 = rsinθ …………eq.2
Squaring and adding eq.1 and eq.2, we get
$2=r^{2}$
Since r is always a positive no., therefore,
$\mathrm{r}=\sqrt{2}$
Hence its modulus is $\sqrt{2}$.
Now, dividing eq.2 by eq.1 , we get,
$\frac{r \sin \theta}{r \cos \theta}=\frac{1}{-1}$
Tanθ = -1
Since $\cos \theta=-\frac{1}{\sqrt{2}}, \sin \theta=\frac{1}{\sqrt{2}}$ and $\tan \theta=-1$. Therefore the $\theta$ lies in second quadrant.
$\operatorname{Tan} \theta=-1$, therefore $\theta=\frac{3 \pi}{4}$
Representing the complex no. in its polar form will be
$\mathrm{Z}=\sqrt{2}\left\{\cos \left(\frac{3 \pi}{4}\right)+i \sin \left(\frac{3 \pi}{4}\right)_{\}}\right.$