Question

Write the argument of $(1+\sqrt{3})(1+i)(\cos \theta +i\sin \theta )$.

Answer 1

we know that properties of,

$\begin{array}{c}(z_1.z_2)=\left(z_1\right)+\left(z_2\right)\\ \Rightarrow (z_1.z_2.z_3)=\left(z_1\right)+\left(z_2\right)+arg\left(z_3\right)\\ \Rightarrow (1+i\sqrt{3})(1+i)(cos\theta +isin\theta )=(1+i\sqrt{3})+arg(1+i)+arg(cos\theta +isin\theta )\\ \\ Now,\\ argumentfindindividually,\\ z_1=1+i\sqrt{3}[\tan \alpha =\left|\frac{Im\left(z_1\right)}{Re\left(z_1\right)}\right|=\left|\frac{\sqrt{3}}{1}\right|=\sqrt{3},\therefore \alpha =\frac{\pi }{6}\\ \therefore \left(z_1\right)=\theta =\alpha =\frac{\pi }{6}\\ z_2=1+i[\tan \alpha =\frac{1}{1}=1\Rightarrow \alpha =\frac{\pi }{4}\\ \therefore \left(z_2\right)=\theta =\alpha =\frac{\pi }{4}\\ z_3=1(\cos \theta +i\sin \theta )\\ \therefore \left(z_3\right)=\theta [\therefore r(\cos \theta +i\sin \theta )=(z)=\theta ,\left|z\right|=\theta \\ \\ Substituteinequationof,\\ (1+i\sqrt{3})(1+i)(cos\theta +isin\theta )=\frac{\pi }{6}+\frac{\pi }{4}+\theta \\ \left(z\right)=\frac{5\pi }{12}+\theta \\ therefor\left(z\right)=\frac{5\pi }{12}+\theta \\ So,thatthelastcomplexis\frac{5\pi }{12}+\theta \end{array}$