document.write( "Question 1209455: find value of: cos(2π/7)+cos(4π/7)+cos(6π/7) \n" ); document.write( "
Algebra.Com's Answer #848865 by ikleyn(53521)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "find value of: cos(2π/7)+cos(4π/7)+cos(6π/7)
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\n" ); document.write( "\n" ); document.write( "            This is a nice problem of the level slightly higher than ordinary school assignment,\r
\n" ); document.write( "\n" ); document.write( "            and its beauty is four times greater than anything you've ever seen before.\r
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document.write( "Consider the doubled sum\r\n" );
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document.write( "    2S = 2cos(2π/7) + 2cos(4π/7) + 2cos(6π/7).\r\n" );
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document.write( "It is nothing else as the sum of six complex numbers\r\n" );
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document.write( "    2S =   (cos(2π/7)  + isin(2π/7))  + (cos(4π/7)  + isin(6π/7))  + (cos(6π/7) + isin(6π/7)) + \r\n" );
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document.write( "         + (cos(12π/7) + isin(12π/7)) + (cos(10π/7) + isin(10π/7)) + (cos(8π/7) + isin(8π/7)).    (2)\r\n" );
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document.write( "( think one minute - why it is so.  I leave it to you. Recall the signs of sine and cosine ! ).\r\n" );
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document.write( "Consider 1 + 2S.  This expression, 1 + 2S,  represents the sum of all complex roots of equation\r\n" );
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document.write( "    z^7 - 1 = 0,   (or  z^7 = 1, which is the same).     (3)\r\n" );
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document.write( "According to Vieta's theorem, the sum of the roots of this equation is equal to the coefficient at z^6\r\n" );
document.write( "in equation (3), which is zero.  So, we conclude that \r\n" );
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document.write( "    1 + 2S = 0,   or   S = -1/2.\r\n" );
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document.write( "At this point, the problem is solved completely.\r\n" );
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document.write( "ANSWER.  cos(2π/7)+cos(4π/7)+cos(6π/7) = -1/2.\r\n" );
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document.write( "You can check it using your calculator .\r\n" );
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\n" ); document.write( "\n" ); document.write( "In regular school, they do not teach such tricks - only at Math circles or Math schools.\r
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\n" ); document.write( "\n" ); document.write( "Or in special Math textbooks - collections of problems for Math circles and Math Olympiads.\r
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\n" ); document.write( "\n" ); document.write( "Or in web-sites like this one.\r
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\n" ); document.write( "\n" ); document.write( "Tutor @math_tutor in his post re-told my solution in his own words.\r
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\n" ); document.write( "\n" ); document.write( "Thank you for popularizing my ideas.\r
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\n" ); document.write( "\n" ); document.write( "// Actually, I am not very happy, when somebody comes and re-tells my solution/solutions
\n" ); document.write( "in his own words.\r
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\n" ); document.write( "\n" ); document.write( "It is the same as if somebody, let say, @math_tutor2020, comes to re-tell Hemingway prose in his own words.\r
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\n" ); document.write( "\n" ); document.write( "As far as I know, no one in healthy mind has ever tried to retell Hemingway or another writer in their own words -
\n" ); document.write( "it's somehow not accepted in the world. But for some reason it is thriving at this forum, UNFORTUNATELY.\r
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\n" ); document.write( "\n" ); document.write( "        Hello,  @math_tuitor2020,  please remove all your insinuations \r
\n" ); document.write( "\n" ); document.write( "            about me from all your posts   -   WITHOUT  DELAY.\r
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\n" ); document.write( "\n" ); document.write( "                                                    Jan. 20, 2025.\r
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