SOLUTION: Find the 3 cube roots of -i in POLAR FORM: rcis(theta).

Algebra ->  Trigonometry-basics -> SOLUTION: Find the 3 cube roots of -i in POLAR FORM: rcis(theta).      Log On


   



Question 1117284: Find the 3 cube roots of -i in POLAR FORM: rcis(theta).
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
The trigonometric form of the complex number  "-i"  is  cos(270°) + i*sin(270°).

The modulus of  "-i"  is  1,  the argument is  270° = 3pi%2F2.

According to the general theory, there are three complex cube roots of  "-i".  They have the modulus of  root%283%2C1%29 = 1. 
The first cube root has the argument of  90° = pi%2F2,  one third of the argument of  "-i". 
Each next cube root has the argument in  360%5Eo%2F3 = 120° = 2pi%2F3  more than the previous one. 
Thus the tree complex roots are


  1)  cos(90°) + i*sin(90°) = cis(90°) = i; 


  2)  cos(90°+120°) + i*sin(90° + 120°) = cos(210°) + i*sin(210°);


  3)  cos(90°+240°) + i*sin(90° + 240°) = cos(330°) + i*sin(330°).

Solved.

---------------
On complex numbers, there is a bunch of lessons in this site
    - 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.

Of them, the most relevant lessons are marked (*) in the list.

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.