This is a free lesson from our course in Trigonometry In this lesson you'll learn with the help of some examples and practice questions with solution, how to find roots of complex numbers. For complex number z = r (cos + i sin ), then for a positive integer n, nth root is given by        n r[cos {( + 2 k)/n} + i sin {( + 2 k)/n}] for values of k from 0, 1, 2, ......, n - 1. For example, if you are asked to find fourth roots of the complex number z = 5(cos /3 + i sin /3), all you need to do is to plug values into the formula. Here, n = 4, r = 5, = /3. Since n is 4, use the values of k = 0, 1, 2, 3. (More text below video...)
Other useful lessons:
 Getting Started: De Moivre's Theorem Using De Moivre's Theorem

(Continued from above) The roots are:
4 5(cos /12 + i sin /12)
4 5(cos 7 /12 + i sin 7 /12)
4 5(cos 13 /12 + i sin 13 /12)
4 5(cos 19 /12 + i sin 19 /12)
Similarly, to find cube roots of -1, first you write it in the polar form and then plug the values in formula three times. The roots are (1/2) + ( 3/2)i, -1, and (1/2) - ( 3/2)i. Once you go through the explanation by instructor in the video, it'll be easy for you to understand how the roots of complex numbers determined.

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