Question

Write the complex number $z=\frac{i-1}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$ in the polar form.

Answer 1

we have, $z=\frac{i-1}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$

$Let{z}_1=i-1\Rightarrow z_1=-1+i$

$|z_1|=\sqrt{(-1{)}^2+(1{)}^2}=\sqrt{2}$

$andtan\alpha =\left|\frac{1}{-1}\right|=1\Rightarrow \alpha =\frac{\pi }{4}$

$\theta =arg\left(z\right)=\pi -\frac{\pi }{4}=\frac{3\pi }{4}$

$[\because x<0,y>o,\theta lieinIIquadrant]$

$\therefore z_1=\sqrt{2}(cos\frac{3\pi }{4}+isin\frac{3\pi }{4})$

$Now,z=\frac{\sqrt{2}(cos\frac{3\pi }{4}+isin\frac{3\pi }{4})}{\sqrt{2}(cos\frac{\pi }{4}+isin\frac{\pi }{4})}$

$=\frac{\sqrt{2}(cos\frac{3\pi }{4}+isin\frac{3\pi }{4})(cos\frac{\pi }{3}-isin\frac{\pi }{3})}{(cos\frac{\pi }{3}+isin\frac{\pi }{3})(cos\frac{\pi }{3}-isin\frac{\pi }{3})}$

$[multipyingnumeratoranddenominatorby\left(cos\frac{\pi }{3}\right)-isin\frac{\pi }{3}]$

$\Rightarrow z=\frac{\sqrt{2}[cos(\frac{3\pi }{4}-\frac{\pi }{3})+isin(\frac{3\pi }{4}-\frac{\pi }{3})]}{co{s}^2\frac{\pi }{3}+so{m}^2\frac{\pi }{3}}$

$\Rightarrow z=\sqrt{2}(cos\frac{5\pi }{12}+isin\frac{5\pi }{12})$

Hence, the polar form of $z=\frac{i-1}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$

is $\sqrt{2}(cos\frac{5\pi }{3}+isin\frac{\pi }{3})$