document.write( "Question 207423: Hi there... i just have a few questions that i would like to ask regarding complex numbers.\r
\n" ); document.write( "\n" ); document.write( "1) $\"1-sqrt%283%29%2Ai\"$
\n" ); document.write( "i would show you how much i can do on this one... however, im completely stuck sorry
\n" ); document.write( "and i was unable to find a proper square root sign, so hope that its alright.\r
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\n" ); document.write( "\n" ); document.write( "2) $\"c+=+sqrt%282%29-i%2Asqrt%282%29\"$ \r
\n" ); document.write( "\n" ); document.write( "I REALLY strggle with De Moivre's theorem.
\n" ); document.write( "for this question i have to find the following:
\n" ); document.write( "a) $\"abs%28c%29\"$
\n" ); document.write( "b)find $\"arg%28c%29\"$ when $\"-pi%3Carg%28c%29%3Cpi\"$
\n" ); document.write( "c)write $\"c\"$ in polar form
\n" ); document.write( "d}find $\"c%5E6\"$ or $\"%28sqrt%282%29+-+i%2Asqrt%282%29%29%5E6\"$ using De Moivre's theorem.\r
\n" ); document.write( "\n" ); document.write( "SO those are all my questions, hope they are alright\r
\n" ); document.write( "\n" ); document.write( "Thanks
\n" ); document.write( "Christelle :D \n" ); document.write( "

Algebra.Com's Answer #156866 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "I'll just do (2) and you can use it as a model to\r\n" );
document.write( "do (1). \r\n" );
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document.write( "Let's draw the picture of that complex number as a\r\n" );
document.write( "right triangle on a graph. \r\n" );
document.write( "where , \r\n" );
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\n" ); document.write( "(a) $\"abs%28c%29\"$ .
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document.write( "Sometimes this is called the \"modulus\". This is the length of \r\n" );
document.write( "the hypotenuse of that right triangle. We find it the same way \r\n" );
document.write( "we always find the hypotenuse, the Pythagorean theorem:\r\n" );
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document.write( "We label the hypotenuse  and this is the same as \r\n" );
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\n" ); document.write( "(b) find $\"arg%28c%29\"$ when $\"-pi%3Carg%28c%29%3Cpi\"$
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document.write( "\"Argument\" means \"angle\" (you can remember it because \"argument\"\r\n" );
document.write( "and \"angle\" both start with \"a\" and their third letters are \"g\")\r\n" );
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document.write( "Now since we are told that the argument  is between \r\n" );
document.write( "and , we must take the argument as a negative angle \r\n" );
document.write( "measured by rotating clockwise from the right side of the x-axis, \r\n" );
document.write( "indicated by the blue arc below labeled , \"phi\".\r\n" );
document.write( "It should be q, \"theta\", but I can't \r\n" );
document.write( "get that Greek letter on the notation program for this site, so \r\n" );
document.write( "I'll use  instead.\r\n" );
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document.write( "Since the triangle is an isosceles right triangle, its interior\r\n" );
document.write( "angles are  or  in radians.  However since \r\n" );
document.write( "tells us that the rotation is clockwise from the right\r\n" );
document.write( "side of the x-axis, the angle is taken as negative. Thus we\r\n" );
document.write( "take it as  and we label it:\r\n" );
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\n" ); document.write( "(c) write in polar form:
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document.write( "Since , therefore \r\n" );
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document.write( "and since , therefore \r\n" );
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document.write( "So \r\n" );
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document.write( "and we can factor out  and get:\r\n" );
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document.write( "This is the polar form  of \r\n" );
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document.write( "Therefore since  and \r\n" );
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\n" ); document.write( "(d) $\"c%5E6\"$ or (sqrt(2)-i*sqrt(2))^6}}}
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document.write( "DeMoivre's theorem says:\r\n" );
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document.write( "Therefore:\r\n" );
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document.write( "Since  is coterminal with , we have\r\n" );
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document.write( "Edwin

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