SOLUTION: Find the fourth roots of 16(cos 200° + i sin 200°).

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Question 1134034: Find the fourth roots of 16(cos 200° + i sin 200°).
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the fourth roots of 16(cos 200° + i sin 200°).
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Find the fourth roots of 16cis(200)
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= 2cis(50)
360/4 = 90
Add 90, then add 90 again, then a 3rd time.

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


deMoivre's Theorem (unfortunately not in the curriculum in many high schools....):



In your problem, a=16, b=200, n = 1/4, so



A result of deMoivre's Theorem is that the n n-th roots of any complex number all have the same magnitude and are distributed around a circle in increments of 360/n degrees. So the four 4th roots of your number are

2%2A%28cos%2850%29%2Bi%2Asin%2850%29%29
2%2A%28cos%28140%29%2Bi%2Asin%28140%29%29
2%2A%28cos%28230%29%2Bi%2Asin%28230%29%29
2%2A%28cos%28320%29%2Bi%2Asin%28320%29%29

Note we can use deMoivre's Theorem to verify that each of those is a 4th root of the given complex number. For example,