Question 290225: Find all the sixth roots of (12+5i)
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! Find all the sixth roots of (12+5i)
we are gonna use de Moivre's formula
[r * (cosx + isinx)]^n = r^n * (cos(nx) + isin(nx))
e^(ix) = cosx + isinx (Euler's formula)
(e^(ix))^n = e^(inx)
e^(i(nx)) = cos(nx) + isin(nx)
(r*e^(ix))^n = r^n*e^(inx) = r^n * (cos(nx) + isin(nx))
convert 12+5i to polar form and set it to z
r = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13
alpha = tan^(-1)(5/12) =approx. 22.619865 degrees
theta = 180 - alpha =approx. 157.380135
z = 12 + 5i = 13(cos(157.380135) + isin(157.380135))
z^(1/6) = (13(cos(157.380135) + isin(157.380135)))^(1/6)
z^(1/6) = 13^(1/6)[cos(1/6 * (157.380135 + 360k)) + isin(1/6 * (157.380135 + 360k))]
(2pi radians is one revolution it is equal to 360degrees)
k from 0 to 5
z^(1/6) = 1.533406 * [cos(26.2300225 + 60k) + isin(26.2300225 + 60k)]
k from 0 to 5
k=0 --> 1.375506 + 0.677729i
k=1 --> 0.100823 + 1.530088i
k=2 --> -1.274683 + 0.852359i
k=3 --> -1.375506 - 0.677729i
k=4 --> -0.100823 - 1.530088i
k=5 --> 1.274683 - 0.852359i
(calculated these by plugging the formulas into Excel and remembering to convert what is in the parenthesis for cos and sin to radians)
**** REVISITING ABOVE PROBLEM ****
I checked the above answers and found that while they did come out to the right magnitude (r^(1/6) or 13^(1/6) or 1.533406) they when plugged into the Pascal Triangle expansion of (a+bi)^6 which is:
a^6 - 15a^4b^2 + 15a^2b^4 - b^6 for the real (a) terms
and 6a^5b - 20a^3b^3 + 6ab^5 for the imaginary (bi) terms, give:
-12 + 5i and not 12 + 5i
I did everything above correct but used the wrong theta, it was a mistake to of subtracted theta from pi radians (180 degrees) when solving above.
The correct theta since a and b are greater than 0 is 0.394791 radians or
22.619865 degrees.
plugging in:
k a bi mag
0 1.530088 0.100823i 1.533406
1 0.677729 1.375507i 1.533406
2 -0.852359 1.274684i 1.533406
3 -1.530088 -0.100823i 1.533406
4 -0.677729 -1.375507i 1.533406
5 0.852359 -1.274684i 1.533406
these all give 12 + 5i
**** I apologize for the inconvenience this may of caused ****
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