Question

The amplitude and modulus of the complex number $-2+2\sqrt{3}i.$ is $4$ and $\frac{\pi }{3}$.

• A. True
• B. False

Answer 1

The correct option is B False

The given complex number is $-2+2\sqrt{3}i$.

The modulus of $-2+2\sqrt{3}i=\sqrt{(-2{)}^2+(2\sqrt{3}{)}^2}=\sqrt{4+12}=\sqrt{16}=4$.

Therefore, the modulus of $-2+2\sqrt{3}i=4$

Clearly, in the z-plane the point $z=-2+2\sqrt{3}i=(-2,2\sqrt{3})$ lies in the second quadrant. Hence, if $ampz=\theta$ then,

$tan\theta =\left(2\sqrt{3}\right)/-2=-\sqrt{3}$ where, $\pi /2<\theta \le \pi$.

Therefore, $tan\theta =-\sqrt{3}=tan(\pi -\frac{\pi }{3})=tan\left[\frac{2\pi )}{3}\right]$

Therefore, $\theta =\frac{2\pi }{3}$

Therefore, the required amplitude of $-2+2\sqrt{3}i$ is $\frac{2\pi }{3}.$