1. ## De moivres theorem.

Use DMT to solve the following equations in polar form.

$z^3=-1-i$
$z^3=-4+4\sqrt{3i}$

2. What have you tried so far?. These problems are just routine knowing the corresponding theory.

3. I'll show you how to do one, then you try to do the other...

$\displaystyle |-1-i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$.

Note that this complex number is in the third quadrant. The angle in the first quadrant is $\displaystyle \arctan{\frac{1}{1}} = \frac{\pi}{4}$.
So the angle in the third quadrant, $\displaystyle \arg{(-1 - i)} = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$.

$\displaystyle z^3 = \sqrt{2}\,\textrm{cis}\,{\left(\frac{5\pi}{4}\righ t)}$

$\displaystyle z = (\sqrt{2})^{\frac{1}{3}}\,\textrm{cis}\,{\left(\fr ac{1}{3}\cdot \frac{5\pi}{4}\right)}$

$\displaystyle = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12}\right)}$.

There are always three cube roots, and they are of the same magnitude and evenly spaced about a circle. So they differ by an angle of $\displaystyle \frac{2\pi}{3}$.

So the other solutions are $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12}\right)}$

and $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{21\pi}{12}\right)}$.

But since $\displaystyle -\pi < \arg{z} \leq \pi$, we rewrite this as

$\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(-\frac{11\pi}{12}\right)}$ and $\displaystyle \sqrt[6]{2}\,{\left(-\frac{3\pi}{12}\right)}$.
The text passage contains a set of mathematical problems related to De Moivre's Theorem, followed by a detailed solution for one of the problems. The solution includes the steps to find the cube roots of a complex number in polar form.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## De moivres theorem.

Use DMT to solve the following equations in polar form.

$z^3=-1-i$
$z^3=-4+4\sqrt{3i}$

2. What have you tried so far?. These problems are just routine knowing the corresponding theory.

3. I'll show you how to do one, then you try to do the other...

$\displaystyle |-1-i| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}$.

Note that this complex number is in the third quadrant. The angle in the first quadrant is $\displaystyle \arctan{\frac{1}{1}} = \frac{\pi}{4}$.
So the angle in the third quadrant, $\displaystyle \arg{(-1 - i)} = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$.

$\displaystyle z^3 = \sqrt{2}\,\textrm{cis}\,{\left(\frac{5\pi}{4}\righ t)}$

$\displaystyle z = (\sqrt{2})^{\frac{1}{3}}\,\textrm{cis}\,{\left(\fr ac{1}{3}\cdot \frac{5\pi}{4}\right)}$

$\displaystyle = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12}\right)}$.

There are always three cube roots, and they are of the same magnitude and evenly spaced about a circle. So they differ by an angle of $\displaystyle \frac{2\pi}{3}$.

So the other solutions are $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{5\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12}\right)}$

and $\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{13\pi}{12} + \frac{2\pi}{3}\right)} = \sqrt[6]{2}\,\textrm{cis}\,{\left(\frac{21\pi}{12}\right)}$.

But since $\displaystyle -\pi < \arg{z} \leq \pi$, we rewrite this as

$\displaystyle \sqrt[6]{2}\,\textrm{cis}\,{\left(-\frac{11\pi}{12}\right)}$ and $\displaystyle \sqrt[6]{2}\,{\left(-\frac{3\pi}{12}\right)}$.
user
The text passage contains a set of mathematical problems related to De Moivre's Theorem, followed by a detailed solution for one of the problems. The solution includes the steps to find the cube roots of a complex number in polar form.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS