Question

Find the modulus and amplitude of $-2+2\sqrt{3}i$

• A. $|z|=2\sqrt[2]{22};amp\left(z\right)=\frac{\pi }{3}$
• B. $|z|=2\sqrt[2]{22};amp\left(z\right)=\frac{2\pi }{3}$
• C. $|z|=4;amp\left(z\right)=\frac{2\pi }{3}$
• D. $|z|=4;amp\left(z\right)=\frac{\pi }{3}$

Answer 1

The correct option is C $|z|=4;amp\left(z\right)=\frac{2\pi }{3}$

Given $z=-2+2\sqrt{3}i$

Hence, $|z|=2\sqrt{1+3}\Rightarrow 2\sqrt{4}$

$\Rightarrow 2\times 2=4$
And let $\alpha$ be the argument of the complex number.
Then
$tan\alpha =-\sqrt{3}$

Now $Re\left(z\right)<0$ and $Im\left(z\right)>0$

$\therefore$ the number lies in the II quadrant
Hence,

$\alpha =\pi -\frac{\pi }{3}$

$=\frac{2\pi }{3}$