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Use de Moivre's theorem to find the three cube roots of -64i. Write your answers in both polar form and a + bi form.
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de Moivre is the name of the French mathematician (1667-1754).
You can read about him from this Wikipedia article https://en.wikipedia.org/wiki/Abraham_de_Moivre
As any/every person's name, it should be written using first CAPITAL letter.
The trigonometric form (= the polar form) of the complex number "-64i" is 64*(cos(270°) + i*sin(270°)).
The modulus of "-64i" is 64, the argument is 270° = .
According to the general theory, there are three complex cube roots of "-64i". They have the modulus of = 4.
The first cube root has the argument of 90° = , one third of the argument of "-64i".
Each next cube root has the argument by = 120° = more than the previous one.
Thus the tree complex roots are
1) 4*(cos(90°) + i*sin(90°)) = ;
2) 4*(cos(90°+120°) + i*sin(90° + 120°)) = 4*(cos(210°) + i*sin(210°) = = .
3) 4*(cos(90°+240°) + i*sin(90° + 240°)) = 4*(cos(330°) + i*sin(330°)) = = .
Solved.
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There is a bunch of my lessons on complex numbers
- Complex numbers and arithmetical operations on them
- Complex plane
- Addition and subtraction of complex numbers in complex plane
- Multiplication and division of complex numbers in complex plane
- Raising a complex number to an integer power
- How to take a root of a complex number (*)
- Solution of the quadratic equation with real coefficients on complex domain
- How to take a square root of a complex number
- Solution of the quadratic equation with complex coefficients on complex domain
- Solved problems on taking roots of complex numbers (*)
- Solved problems on arithmetic operations on complex numbers
- Solved problem on taking square root of complex number
- Miscellaneous problems on complex numbers
- Advanced problem on complex numbers
- Solved problems on de'Moivre formula
- A curious example of an equation in complex numbers which HAS NO a solution
in this site.
In this list, I marked by (*) the lessons that are most relevant to this problem.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Complex numbers".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.