Question

Find the amplitude and modulus of the complex number -2 + 2√3i

Answer 1

Answer: 4

Step-by-step explanation:

The amplitude and modulus of the complex number -2 + 2√3i is

= √( (-2)^2 + (2√3)^2 )

= √(4 + 12)

= √16

= 4

Answer 2

Answer:

The given complex number is

$- 2 + \sqrt{3i}$

The modulus of

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

Therefore, the modulus of

$- 2 + \sqrt{3i } = 4$

clearly , in the the z-plane the point

$z = - 2 + \sqrt{3i} = (- 2 ,2 \sqrt{3} )$

lies in the second quadrant Hence, if ampz=θ then ,

$\tan( θ) = \frac{(2 \sqrt{3} )}{ - 2} = - \sqrt{3} \: where \: \frac{\pi}{2} <θ \leqslant \pi$

Therefore,

$\tan( θ ) = - \sqrt{3} = \tan(\pi - \frac{\pi}{3} ) = \tan( \frac{2\pi}{3} )$

Therefore,

$θ = \frac{2\pi}{3}$

•°• the required amplitude of

$- 2 + 2 \sqrt{3i} \: \: \: \: \: is \: \: \: \: \: \frac{2\pi}{3}$