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Nugget
Mathematics
28 December, 02:29
The complex fourth roots of 5-5sqrt3i
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Camren Nielsen
28 December, 02:44
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First convert the complex number to polar form
|5 - 5sqrt3i| = 25sqrt10
argument = arctan - sqrt3 = - pi/3
so 5 - 5sqrt3i = 25sqrt10 (cos (-pi/3) + i sin (-pi/3))
the angle is in the 4th quadrant so we could write it as 2pi-pi/3 = 5p/3
= 25sqrt10 (cos 5pi/3 + i sin 5pi/3)
Now if r (cosx + isin x) is a 5th root of 5-5sqrt3i then
r^5 (cos x + i sinx) ^5 = 25sqrt10 (cos 5pi/3 + i sin 5pi/3)
r^5 = 25sqrt10 and cos5x + i sin 5x = cos 5pi/3 + i sin 5pi/3
i have to go urgently so i have to leave it to you to finish this
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