The amplitude of the complex number $\frac{1+i\sqrt{3}}{\sqrt{3}+i}$ is

Solution

1. Let $z_1=1+i\sqrt{3}$ and $z_2=\sqrt{3}+i$.
2. The amplitude of $z_1$ is $$\begin{array}{rl}\text{arg}(1+i\sqrt{3})& ={\tan }^{-1}(\sqrt{3})\\ & =\frac{\pi }{3}\end{array}$$.
3. The amplitude of $z_2$ is $$\begin{array}{rl}\text{arg}(\sqrt{3}+i)& ={\tan }^{-1}(\frac{1}{\sqrt{3}})\\ & =\frac{\pi }{6}\end{array}$$.
4. Using the property $\text{arg}(\frac{z_1}{z_2})=\text{arg}(z_1)-\text{arg}(z_2)$, we calculate $\frac{\pi }{3}-\frac{\pi }{6}=\frac{\pi }{6}$.
5. Hence the answer is (A).