document.write( "Question 1152856: Find an expression for cos(7A) and hence Solve: \"P%28x%29+=+64x%5E6+%2B+64x%5E5+-+48x%5E4+-+48x%5E3+%2B+8x%5E2+%2B+8x+%2B+1\"
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Algebra.Com's Answer #774965 by ikleyn(52915)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            The idea and the direction are THESE :\r
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document.write( "Consider complex number  \r\n" );
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document.write( "    z = cos(a) + i*sin(a).\r\n" );
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document.write( "Raise in degree 7.  You will get\r\n" );
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document.write( "    \"z%5E7\" = \"%28cos%28a%29+%2B+i%2Asin%28a%29%29%5E7\".\r\n" );
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document.write( "Left side, due to de Moivre formula, is  cos(7a) + i*sin(7a).  So, you have\r\n" );
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document.write( "    cos(7a) + i*sin(7a) = \"%28cos%28a%29+%2B+i%2Asin%28a%29%29%5E7\".\r\n" );
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document.write( "In the right side, decompose using the Newton's binomial formula.\r\n" );
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document.write( "On the way, use identities  \"i%5E2\" = -1;  \"i%5E3\" = -i;  \"i%5E4\" = 1 and so on.\r\n" );
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document.write( "Next, separate real and imaginary parts.  From real part, you will get the expression for cos(7a) via cos(a) and sin(a);\r\n" );
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document.write( "in this expression, all sin(a) will be in even degrees.\r\n" );
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document.write( "Exclude sin(a) in even degrees in this expression, using the formula \"sin%5E2%28a%29\" = \"1-cos%5E2%28a%29\".\r\n" );
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document.write( "In this way, you will get the expression\r\n" );
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document.write( "    cos(7a) = f(cos(a)),      (1)\r\n" );
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document.write( "where f(x) is some polynomial of x.\r\n" );
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document.write( "This polynomial is exactly what you are looking for.\r\n" );
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document.write( "Now, to find the root of this polynomial, notice that cos(7a) = 0 if  a = \"pi%2F14\".\r\n" );
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document.write( "Therefore,  x= \"cos%28pi%2F14%29\"  is the root of the polynomial (1).\r\n" );
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document.write( "By the way, its other roots are  \"cos%28%283pi%29%2F14%29\",  \"cos%28%285pi%29%2F14%29\",  \"cos%28%287pi%29%2F14%29\",  \"cos%28%289pi%29%2F14%29\",  \"cos%28%2811pi%29%2F14%29\",  and  \"cos%28%2813pi%29%2F14%29\",  due to the same reason (!)\r\n" );
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\n" ); document.write( "\n" ); document.write( "Similar problem was solved in the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on de'Moivre formula\r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of my lessons on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on de'Moivre formula\r
\n" ); document.write( "\n" ); document.write( "    - Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°) \r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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