Question

Express the following complex numbers in the form $r\left(\cos \theta +i\sin \theta \right)$ :
1 + i tan$\alpha$

Answer 1

$1+i\tan \alpha$

$r=\sqrt{1^2+{\tan }^2\alpha }=\sec \alpha$

$arg\left(\theta \right)={\tan }^{-1}\left(\frac{\tan \alpha }{1}\right)$

$=\alpha ,\alpha >0$

$=-\alpha ,\alpha <0$

The required form is $(\sec \alpha ,\alpha ),(\sec \alpha ,-\alpha )$