Question

asked Sep 25, 2021 178k views

2 Answers

Schadensbegrenzer · Answer 1 · 2021-09-30T12:30:12+0000

Answer:

|z| = 27 -- Modulus

$\theta = (\pi)/(3)$ --- Argument

Explanation:

Given

$((\sqrt 3)(cos(\pi)/(18) + i\ sin(\pi)/(18)))^6$

Required

Determine the modulus and the argument

We have that:

$z = ((\sqrt 3)(cos(\pi)/(18) + i\ sin(\pi)/(18)))^6$

Expand:

$z = (\sqrt 3)^6(cos(\pi)/(18) + i\ sin(\pi)/(18))^6$

$z = 27(cos(\pi)/(18) + i\ sin(\pi)/(18))^6$

A complex equation can be expressed as:

$z = |z| e^(i\theta)$

Where

|z| = modulus

$\theta = argument$

Where

$e^(i\theta) = (cos(\pi)/(18) + i\ sin(\pi)/(18))$

So:

$z = 27(cos(\pi)/(18) + i\ sin(\pi)/(18))^6$ becomes

$z = 27(e^{i(\pi)/(18)})^6$

By comparison:

$e^(i\theta) = (e^{i(\pi)/(18)})^6$

This gives:

${i\theta} = i(\pi)/(18)}*6$

${i\theta} = i(6\pi)/(18)}$

${i\theta} = i(\pi)/(3)}$

Divide through by i

$\theta = (\pi)/(3)$

Hence, the modulus, z is:

|z| = 27

And the argument
$\theta$ is

$\theta = (\pi)/(3)$

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