document.write( "Question 1073428: Find the cube roots of the following complex number;\r
\n" ); document.write( "\n" ); document.write( " -5(square root of 2)+ 5(square root of 2i)
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Algebra.Com's Answer #688325 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
I believe what you wanted was the cube roots of
\n" ); document.write( "\"-5sqrt%282%29%2Bi5sqrt%282%29\" .
\n" ); document.write( "You can write that as
\n" ); document.write( "-5sqrt(2)+i5sqrt(2) or as -5*sqrt(2)+i*5*sqrt(2) .
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\n" ); document.write( "The three cube roots will be numbers of the form
\n" ); document.write( "\"root%283%2C10%29%28cos%28theta%29%2Bi%2Asin%28theta%29%29\" such that
\n" ); document.write( "\"%28cos%28theta%29%2Bi%2Asin%28theta%29%29%5E3=cos%28135%5Eo%29%2Bi%2Asin%28135%5Eo%29\" ,
\n" ); document.write( "meaning that \"3theta=135%5Eo%2Bk%2A360%5Eo\" for
\n" ); document.write( "\"system%28k=0%2Ck=1%2C%22or%22%2Ck=2%29\" .
\n" ); document.write( "So, \"theta=135%5Eo%2F3%2Bk%2A120%5Eo\" gives us
\n" ); document.write( "\"theta=45%5Eo\" , \"theta=165%5Eo\" and \"theta=285%5Eo\" .
\n" ); document.write( "You can get approximate values for \"root%283%2C10%29\"
\n" ); document.write( "and for the trigonometric functions from a calculator.
\n" ); document.write( "\"root%283%2C10%29%28cos%2845%5Eo%29%2Bi%2Asin%2845%5Eo%29%29=approximately\"\"highlight%281.523%2B1.523i%29\"
\n" ); document.write( "\"root%283%2C10%29%28cos%28165%5Eo%29%2Bi%2Asin%28165%5Eo%29%29=approximately\"\"highlight%28-2.081%2B0.558i%29\"
\n" ); document.write( "\"root%283%2C10%29%28cos%28285%5Eo%29%2Bi%2Asin%28285%5Eo%29%29=approximately\"\"highlight%280.558-2.081i%29\"
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\n" ); document.write( "Finding exact values is a little more involved,
\n" ); document.write( "and the resulting expressions may not look as nice as with approximate decimals.
\n" ); document.write( "The one answer with \"theta=45%5Eo\" is easy,
\n" ); document.write( "because \"cos%2845%5Eo%29=sin%2845%5Eo%29=sqrt%282%29%2F2\" ,
\n" ); document.write( "so that the exact value for one of the three cube roots is
\n" ); document.write( " .
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\n" ); document.write( "For the other two values of \"theta\" , it gets a little more complicated.
\n" ); document.write( "Since \"165%5Eo=180%5Eo-15%5Eo\" and \"285%5Eo=270%5Eo%2B15%5Eo\" ,
\n" ); document.write( "we can use the trigonometric functions for \"15%5Eo\" ,
\n" ); document.write( "knowing their relation to the trigonometric functions for \"165%5Eo\" and \"285%5Eo\" .
\n" ); document.write( "For second quadrant \"theta=165%5Eo\" ,
\n" ); document.write( "\"cos%28165%5Eo%29=-cos%2815%5Eo%29\" and \"sin%28165%5Eo%29=sin%2815%5Eo%29\" .
\n" ); document.write( "For fourth quadrant \"theta=285%5Eo\" ,
\n" ); document.write( "\"cos%28285%5Eo%29=-sin%2815%5Eo%29\" and \"sin%28285%5Eo%29=cos%2815%5Eo%29\" .
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\n" ); document.write( "If we want exact values, we will need to use the half-angle trigonometric formulas:
\n" ); document.write( "\"cos%5E2%28A%2F2%29=%281%2Bcos%28A%29%29%2F2\" and \"sin%5E2%28A%2F2%29=%281-cos%28A%29%29%2F2\" .
\n" ); document.write( "\"cos%2830%5Eo%29=sqrt%283%29%2F2\" , so
\n" ); document.write( " and .
\n" ); document.write( "So, \"cos%2815%5Eo%29=sqrt%282%2Bsqrt%283%29%29%2F2\" and \"sin%2815%5Eo%29=sqrt%282-sqrt%283%29%29%2F2\"
\n" ); document.write( "That makes the exact values for the other two cube roots
\n" ); document.write( "\"root%283%2C10%29%28-sqrt%282%2Bsqrt%283%29%29%2F2%2Bi%28sqrt%282-sqrt%283%29%29%2F2%29%29\"= and
\n" ); document.write( "\"root%283%2C10%29%28sqrt%282-sqrt%283%29%29%2F2%29-i%28sqrt%282%2Bsqrt%283%29%29%2F2%29%29\"= . \n" ); document.write( "
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