document.write( "Question 981152: Find the cube roots of 27(cos 279° + i sin 279°).\r
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\n" ); document.write( "\n" ); document.write( "(27*(cos 279 + isin279)^1/3= 27^1/3 * (cos 279 + i sin 279) ^ 1/3
\n" ); document.write( "27 ^ 1/3 = cube root 27 = 3
\n" ); document.write( "3*(cos 279 + i sin 279) ^1/3= 3(cos 279/3 + i sin 279/3)= 3(cos 93 + i sin 93) \r
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Algebra.Com's Answer #602178 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Notice that in the instructions the word \"roots\" is plural.\r\n" );
document.write( "In complex numbers there are always \"n nth roots\" of any \r\n" );
document.write( "complex number except 0. \r\n" );
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document.write( "As the other tutor pointed out, you have found only one cube root.  \r\n" );
document.write( "However there are 3 cube roots.  You must begin by adding 360°k \r\n" );
document.write( "to the angle since the sines and cosines are the same when we \r\n" );
document.write( "add any multiple of 360° to the angle.\r\n" );
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document.write( "must first be written as\r\n" );
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document.write( "Then we do what you did.  We take the real 1/3 power (or \r\n" );
document.write( "cube root) of 27 and multiply the angle by \"1%2F3\"\r\n" );
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document.write( "Now we choose k as any three consecutive integers, say 0,1,and 2.\r\n" );
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document.write( "One cube root will be found by substituting k=0\r\n" );
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document.write( "Cube root #1:    \"3%28cos%28%2293%B0%22%29%5E%22%22%2Bi%2Asin%28%2293%B0%22%29%29%29%29\", which was the only one you found.\r\n" );
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document.write( "Use k=1 (which amounts to adding 120° to 93°\r\n" );
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document.write( "The second cube root will be found by substituting k=1\r\n" );
document.write( "[Using k=1 amounts to adding 120° to 93°]\r\n" );
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document.write( "cube root #2:   \"3%28cos%28%22213%B0%22%29%5E%22%22%2Bi%2Asin%28%22213%B0%22%29%29%29%29\",\r\n" );
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document.write( "Use k=2 (which amounts to adding 120° to 213°\r\n" );
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document.write( "The third and final cube root will be found by substituting k=2\r\n" );
document.write( "[Using k=2 amounts to adding 120° to 213°]\r\n" );
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document.write( "cube root #3:   \"3%28cos%28%22333%B0%22%29%5E%22%22%2Bi%2Asin%28%22333%B0%22%29%29%29%29\",\r\n" );
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document.write( "If we plot those cube roots on a graph, where the real part\r\n" );
document.write( "is the x-coordinate and the imaginary part is the y-coordinate,\r\n" );
document.write( "we find that they are three vectors each 3 units long and equally \r\n" );
document.write( "spaced around a circle of radius 3 like spokes of a wheel.  \r\n" );
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document.write( "Edwin
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