document.write( "Question 1180531: Find the 3 cube roots of i in polar form. \n" ); document.write( "
Algebra.Com's Answer #810239 by Edwin McCravy(20060)\"\" \"About 
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document.write( "While correct, neither the notation (r,θ) nor rei*θ is used when teaching complex numbers \r\n" );
document.write( "in basic trigonometry courses.  That's because such notation eliminates the trigonometric functions.\r\n" );
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Find the 3 cube roots of i in polar form.
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document.write( "i = 0 + 1i\r\n" );
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document.write( "Graph the vector whose magnitude (modulus) is r=1, whose tail is at (0,0),\r\n" );
document.write( "and whose tip is at (0,1), and whose argument (angle) is θ=90o. \r\n" );
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document.write( "\"i\"\"%22%22=%22%22\"\"0%2B1%2Ai\"\"%22%22=%22%22\"\"r%28cos%28theta%29%5E%22%22%2Bi%2Asin%28theta%29%29\"\"1%28cos%2890%5Eo%29%5E%22%22%2Bi%2Asin%2890%5Eo%29%29\"\r\n" );
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document.write( "Since the cube root is the 1/3 power:\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ci%5E%281%2F3%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\r\n" );
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document.write( "We raise everything to the 1/3 power.  In doing so we will use deMoivre's\r\n" );
document.write( "theorem, where we raise the magnitude (modulus 1) to the 1/3 power (i.e.,\r\n" );
document.write( "take its cube root 1), and multiply its argument (angle) by 1/3.\r\n" );
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document.write( "Now, since there are 3 cube roots, we take three consecutive integers for n.\r\n" );
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document.write( "Let n=0\r\n" );
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document.write( "\"1%28cos%2830%5Eo%2B120%5Eo%2A0%29%5E%22%22%2Bi%2Asin%2830%5Eo%2B120%5Eo%2A0%29%29\"\"%22%22=%22%22\"\"1%28cos%2830%5Eo%29%2Bi%2Asin%2830%5Eo%29%5E%22%22%29\"\r\n" );
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document.write( "Let n=1\r\n" );
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document.write( "\"1%28cos%2830%5Eo%2B120%5Eo%2A1%29%5E%22%22%2Bi%2Asin%2830%5Eo%2B120%5Eo%2A1%29%29\"\"%22%22=%22%22\"\"1%28cos%28150%5Eo%29%2Bi%2Asin%28150%5Eo%29%5E%22%22%29\"\r\n" );
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document.write( "Let n=2\r\n" );
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document.write( "\"1%28cos%2830%5Eo%2B120%5Eo%2A2%29%5E%22%22%2Bi%2Asin%2830%5Eo%2B120%5Eo%2A2%29%29\"\"%22%22=%22%22\"\"1%28cos%2830%5Eo%2B240%5Eo%29%2Bi%2Asin%2830%5Eo%2B240%5Eo%29%5E%22%22%29\"\"%22%22=%22%22\"\"1%28cos%28270%5Eo%29%2Bi%2Asin%28270%5Eo%29%5E%22%22%29\".\r\n" );
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document.write( "Edwin
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