Question

how to find modulus and amplitude of √3-i

Answer 1

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COMPLEX NUMBERS....

Answer 2

Answer:  The required modulus of the given complex number is 2 and its amplitude is $\dfrac{5\pi}{3}.$

Step-by-step explanation:  We are given to explain the method of finding the modulus and amplitude of the following complex number :

$z=\sqrt3-i.$

We know that

for a complex number z = a + bi, the modulus and amplitude is given by

$|z|=\sqrt{a^2+b^2},~~~~~\theta=\tan^{-1}\dfrac{b}{a}.$

So, the modulus of the given complex number is

$|z|=\sqrt{(\sqrt3)^2+(-1)^2}=\sqrt{3+1}=\sqrt4=2$

and the amplitude is given by

$\theta=\tan^{-1}\left(-\dfrac{1}{\sqrt3}\right)=\tan^{-1}\left(-\tan\dfrac{\pi}{6}\right)=\tan^{-1}\tan\left(\pi-\dfrac{\pi}{6}\right)\\\\\\\Rightarrow \theta=\dfrac{5\pi}{3}.$

Thus, the required modulus of the given complex number is 2 and its amplitude is $\dfrac{5\pi}{3}.$