document.write( "Question 1208302: z = [1, theta ] , find (z + z ^(10))/(z - z ^(10)) by Polar form [ r , theta ], \n" ); document.write( "

Algebra.Com's Answer #846636 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "z = (1,theta), find (z + z^(10))/(z - z^(10)) $\"highlight%28cross%28by%29%29\"$ in polar form (r,theta).
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document.write( "In cis-form,  z = .   It means that the modulus of z is 1 and the argument of z is .\r\n" );
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document.write( "Then   = .  In other words,   has the modulus 1  and the argument of  .\r\n" );
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document.write( "To find the sum  z + ,  you should use the parallelogram rule of adding complex numbers as vectors.\r\n" );
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document.write( "Since both addends,  z  and    have equal modulus, the parallelogram rule in this case becomes\r\n" );
document.write( "the rhombus rule.  In rhombus, the diagonal is the bisector of the angle.  \r\n" );
document.write( "THEREFORE, in our case the argument of    is\r\n" );
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document.write( "     = . \r\n" );
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document.write( "The modulus of    is the third side of an isosceles triangle with the equal legs of the length 1 \r\n" );
document.write( "and the angle between these legs of   = \r\n" );
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document.write( "    A =  = .\r\n" );
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document.write( "So,   = .\r\n" );
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document.write( "Similarly, for the denominator\r\n" );
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document.write( "     =  = ,\r\n" );
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document.write( "where  B = .   <<<---=== Note that the angle between  z  and    is  \r\n" );
document.write( "                                              = \r\n" );
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document.write( "                                             and   = .\r\n" );
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document.write( "Thus the ratio    has the modulus  \r\n" );
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document.write( "     =    \r\n" );
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document.write( "and the argument as the difference\r\n" );
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document.write( "     =  = .\r\n" );
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document.write( "Using formulas    = ,   = ,  the modulus of the ratio can be simplified\r\n" );
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document.write( "      =  = \r\n" );
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document.write( "Therefore, the ratio    is  ,  or, which is the same, (,).\r\n" );
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document.write( "ANSWER.  In polar form,    is   (,). \r\n" );
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\n" ); document.write( "\n" ); document.write( "The key ideas of solution are \r
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\n" ); document.write( "\n" ); document.write( "     - use cis-form of complex numbers;\r
\n" ); document.write( "\n" ); document.write( "     - use the parallelogram rule to add and to subtract complex numbers;\r
\n" ); document.write( "\n" ); document.write( "     - use the fact that the parallelogram rule in the case of equal modulus becomes the rhombus rule;\r
\n" ); document.write( "\n" ); document.write( "     - use the fact that in rhombus a diagonal bisects the angle of the rhombus\r
\n" ); document.write( "\n" ); document.write( "     - to find the modulus of the numerator and denominator, use the cosine law.
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\n" ); document.write( "\n" ); document.write( "The rest is just arithmetic (quite tight).\r
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\n" ); document.write( "\n" ); document.write( "This problem is special. Its level is much higher than the average high school Math;
\n" ); document.write( "it is higher than teachers teach in Math schools, higher than average Math circles level,
\n" ); document.write( "different from regular Math Olympiads.\r
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\n" ); document.write( "\n" ); document.write( "It is the level of Math Olympiads among undergraduate Math students of Math departments
\n" ); document.write( "of renowned universities/colleges, like the Putnam competition.\r
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\n" ); document.write( "\n" ); document.write( "It does not require \"flight of thought\", but requires solid and firm knowledge of complex numbers
\n" ); document.write( "in all relevant aspects and firm knowledge of relevant adjacent Math subjects,
\n" ); document.write( "together with perfect and firm Math technique.\r
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