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Question

Mathematics Question on complex numbers

Amplitude of 1+3i3+1\frac{1+\sqrt{3}i}{\sqrt{3}+1} is :

A

π6\frac{\pi}{6}

B

π4\frac{\pi}{4}

C

π3\frac{\pi}{3}

D

π2\frac{\pi}{2}

Answer

π3\frac{\pi}{3}

Explanation

Solution

Let r(cos θ+i sin θ)r\left(cos\,\theta+i\,sin\,\theta\right)
=1+i33+1=13+1+i33+1= \frac{1+i\sqrt{3}}{\sqrt{3}+1} =\frac{1}{\sqrt{3}+1}+i\frac{\sqrt{3}}{\sqrt{3}+1}
⇒r cos θ=13+1;r sin θ=33+1\Rightarrow r\,cos\,\theta = \frac{1}{\sqrt{3}+1} ; r\,sin\,\theta = \frac{\sqrt{3}}{\sqrt{3}+1}
⇒tan θ=3⇒θ=π3.\Rightarrow tan\, \theta = \sqrt{3} \Rightarrow \theta = \frac{\pi}{3}.