document.write( "Question 1208305: ((sin (x) + i cos (x)))^9 = sin (9 x) + i cos (9 x) , (true or false \n" ); document.write( "

Algebra.Com's Answer #846656 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "((sin(x) + i*cos(x)))^9 = sin(9x) + i*cos(9x), True or False.
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\n" ); document.write( "\n" ); document.write( " It is a joke Math problem. I will show you a simple elegant solution,\r
\n" ); document.write( "\n" ); document.write( " also in a joking style. Consider it smiling.\r
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document.write( "sin(x) + i*cos(x) is not precisely a standard trigonometry form of a complex number.\r\n" );
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document.write( "As you know, a standard trigonometry form of a complex number is  cos(x) + i*sin(x).\r\n" );
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document.write( "      But we have what is given - therefore, I will apply a trick.\r\n" );
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document.write( "I will write  sin(x) + i*cos(x) = i*(-i*sin(x) + cos(x)) = i*(cos(x) - i*sin(x)).\r\n" );
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document.write( "    Now I will use the fact that cosine is an even function, \r\n" );
document.write( "    while sine is an odd function. Keeping it in mind, I can write\r\n" );
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document.write( "        i*(cos(x) - i*sin(x)) = i*(cos(-x) + i*sin(-x)).\r\n" );
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document.write( "So, now in parentheses I have a standard trigonometry form of complex number  \r\n" );
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document.write( "        cos(-x) + i*sin(-x)  for argument -x.\r\n" );
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document.write( "OK, very good.  Now I raise  sin(x) + i*cos(x), which is given to me,  \r\n" );
document.write( "to degree 9.  Keeping in mind everything above, I have\r\n" );
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document.write( "     =  =  = i*(cos(-9x)+i*sin(-9x)).  (1)\r\n" );
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document.write( "Here I used i^9 = i  and de Moivre formula - standard facts from complex numbers.\r\n" );
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document.write( "    Very good.  Now I will use again that cosine is an even function, \r\n" );
document.write( "    while sine is an odd function.  Keeping it in mind, I can write\r\n" );
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document.write( "        i*(cos(-9x)+i*sin(-9x)) = i*(cos(9x) - i*sin(9x)).    (2)\r\n" );
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document.write( "Last step is to open parentheses in the right side of this formula\r\n" );
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document.write( "        i*(cos(9x) - i*sin(9x) = sin(9x) + i*cos(9x).   (3)\r\n" );
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document.write( "Thus, combining formulas (1), (2) and (3),  you get  \r\n" );
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document.write( "         = sin(9x) + i*cos(9x).\r\n" );
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document.write( "It is what the problem asks to check.\r\n" );
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document.write( "The ANSWER  to the problem's question is True.\r\n" );
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\n" ); document.write( "\n" ); document.write( "As I said at the very beginning, this problem is a Math joke.\r
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\n" ); document.write( "\n" ); document.write( "These tricks provide a complete elegant solution.\r
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\n" ); document.write( "\n" ); document.write( "In a humorous way, if you turn a sock inside out twice,
\n" ); document.write( "it will return to its original form.\r
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\n" ); document.write( "\n" ); document.write( "The meaning of this solution for you is to get understanding
\n" ); document.write( "that this problem is intended to be solved via these tricks with smiling.\r
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\n" ); document.write( "\n" ); document.write( "It is a good problem for a teacher to show it to students at a Math circle.
\n" ); document.write( "First, the students will be shocked seeing these tricks.
\n" ); document.write( "Second, it will help them to master better manipulations with complex numbers.\r
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\n" ); document.write( "\n" ); document.write( "Such problems and similar solutions have a great educational value.
\n" ); document.write( "If students will master them, they will teach next generations in the same way enthusiastically.
\n" ); document.write( "They are like a fuel for the engine of mathematical education, and they are real treasures.\r
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