Question

i-1
5 Convert the complex number
in the polar form
70
TC
tisin​

Answer 1

$\Large{\underline{\sf{\purple{To\:Convert}}}}$

• Convert -1+i into polar form .

$\Large{\underline{\sf{\purple{Solution}}}}$

Let z = -1 + i . Now we have to find out the value of r .

$\sf{\implies r = | z | }$

$\sf{ \implies \: |z| = \sqrt{( {Im(z))}^{2} + ( {Re(z)}^{2} } }$

$\sf{ \implies \: |z| = \sqrt{({ 1)}^{2} + ( {-1})^{2} } }$

• ${\underline{\underline{\sf{ \implies \: |z| = \sqrt{2}}}}}$

Let $\alpha$ be the acuteangle given by

$\sf{ \implies \tan( \alpha ) = | \frac{Im(z)}{Re(z)} | }$

$\sf{ \implies \tan( \alpha ) = | \frac{1}{-1} | }$

$\sf{ \implies \tan( \alpha ) = 1 }$

$\sf{ \implies \alpha = \frac{\pi}{4} }$

The poetry poetry presenting Z lies in the second quadrant . So , the the argument $\theta$ of z is given by

$\sf{ \implies \: \theta \: = \pi - \alpha }$

$\sf{ \implies \: \theta \: = \pi - \frac{\pi}{4} }$

• ${\underline{\underline{\sf{ \implies \: \theta \: = \frac{3 \pi}{4} }}}}$

Hence , thepolar form of Z = -1+i is

$\sf{ \implies \:z = r( \cos( \theta) + i \sin( \theta) ) }$

${\underline{\underline{\sf{\purple{\implies \:z = \sqrt{2} ( \cos( \frac{3 \pi}{4}) + i \sin( \frac{3 \pi}{4}) )}}}}}$