SOLUTION: Find the cube roots of the following complex number; -5(square root of 2)+ 5(square root of 2i)

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Question 1073428: Find the cube roots of the following complex number;
-5(square root of 2)+ 5(square root of 2i)
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Check if you correctly wrote parentheses.


Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I believe what you wanted was the cube roots of
-5sqrt%282%29%2Bi5sqrt%282%29 .
You can write that as
-5sqrt(2)+i5sqrt(2) or as -5*sqrt(2)+i*5*sqrt(2) .


The three cube roots will be numbers of the form
root%283%2C10%29%28cos%28theta%29%2Bi%2Asin%28theta%29%29 such that
%28cos%28theta%29%2Bi%2Asin%28theta%29%29%5E3=cos%28135%5Eo%29%2Bi%2Asin%28135%5Eo%29 ,
meaning that 3theta=135%5Eo%2Bk%2A360%5Eo for
system%28k=0%2Ck=1%2C%22or%22%2Ck=2%29 .
So, theta=135%5Eo%2F3%2Bk%2A120%5Eo gives us
theta=45%5Eo , theta=165%5Eo and theta=285%5Eo .
You can get approximate values for root%283%2C10%29
and for the trigonometric functions from a calculator.
root%283%2C10%29%28cos%2845%5Eo%29%2Bi%2Asin%2845%5Eo%29%29=approximatelyhighlight%281.523%2B1.523i%29
root%283%2C10%29%28cos%28165%5Eo%29%2Bi%2Asin%28165%5Eo%29%29=approximatelyhighlight%28-2.081%2B0.558i%29
root%283%2C10%29%28cos%28285%5Eo%29%2Bi%2Asin%28285%5Eo%29%29=approximatelyhighlight%280.558-2.081i%29

Finding exact values is a little more involved,
and the resulting expressions may not look as nice as with approximate decimals.
The one answer with theta=45%5Eo is easy,
because cos%2845%5Eo%29=sin%2845%5Eo%29=sqrt%282%29%2F2 ,
so that the exact value for one of the three cube roots is
.

For the other two values of theta , it gets a little more complicated.
Since 165%5Eo=180%5Eo-15%5Eo and 285%5Eo=270%5Eo%2B15%5Eo ,
we can use the trigonometric functions for 15%5Eo ,
knowing their relation to the trigonometric functions for 165%5Eo and 285%5Eo .
For second quadrant theta=165%5Eo ,
cos%28165%5Eo%29=-cos%2815%5Eo%29 and sin%28165%5Eo%29=sin%2815%5Eo%29 .
For fourth quadrant theta=285%5Eo ,
cos%28285%5Eo%29=-sin%2815%5Eo%29 and sin%28285%5Eo%29=cos%2815%5Eo%29 .

If we want exact values, we will need to use the half-angle trigonometric formulas:
cos%5E2%28A%2F2%29=%281%2Bcos%28A%29%29%2F2 and sin%5E2%28A%2F2%29=%281-cos%28A%29%29%2F2 .
cos%2830%5Eo%29=sqrt%283%29%2F2 , so
and .
So, cos%2815%5Eo%29=sqrt%282%2Bsqrt%283%29%29%2F2 and sin%2815%5Eo%29=sqrt%282-sqrt%283%29%29%2F2
That makes the exact values for the other two cube roots
root%283%2C10%29%28-sqrt%282%2Bsqrt%283%29%29%2F2%2Bi%28sqrt%282-sqrt%283%29%29%2F2%29%29= and
root%283%2C10%29%28sqrt%282-sqrt%283%29%29%2F2%29-i%28sqrt%282%2Bsqrt%283%29%29%2F2%29%29= .