document.write( "Question 1046276: Prove that:
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Algebra.Com's Answer #661861 by ikleyn(52847) $\"\"$ $\"About$

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\n" ); document.write( "Prove that:
\n" ); document.write( "cos(2pi/7)+cos(4pi/7)+cos(8pi/7)= - 1/2
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document.write( "Imagine the unit circle on a coordinate plane with the center O at the origin of the coordinate system.\r\n" );
document.write( "Imagine that the regular 7-sided polygon ABCDEFG with the center at the point O \r\n" );
document.write( "is inscribed into this circle in a way that one its vertex is located at the point A = (1,0).\r\n" );
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document.write( "Then the other 6 vertices are the points\r\n" );
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document.write( "#2, B: (cos( 2pi/7), sin( 2pi/7))\r\n" );
document.write( "#3, C: (cos( 4pi/7), sin( 4pi/7))\r\n" );
document.write( "#4, D: (cos( 6pi/7), sin( 6pi/7))\r\n" );
document.write( "#5, E: (cos( 8pi/7), sin( 8pi/7))\r\n" );
document.write( "#6, F: (cos(10pi/7), sin(10pi/7))\r\n" );
document.write( "#7, G: (cos(12pi/7), sin(12pi/7))\r\n" );
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document.write( "listed in the anticlockwise order.\r\n" );
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document.write( "1.  First of all, it is clear that the x-coordinates of the points D and E are the same: cos( 6pi/7) = cos( 8pi/7).\r\n" );
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document.write( "    So, we can consider the equal sum  cos(2pi/7)+cos(4pi/7)+cos(6pi/7)  instead  of  cos(2pi/7)+cos(4pi/7)+cos(8pi/7).\r\n" );
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document.write( "2.  From symmetry, it is clear that the sum of vectors  OA + OB + OC + OD + OE + OF + OG  is equal to zero:\r\n" );
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document.write( "    OA + OB + OC + OD + OE + OF + OG = 0.\r\n" );
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document.write( "    It is so obvious that I will omit the proof of this fact.\r\n" );
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document.write( "    It gives us the equality\r\n" );
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document.write( "    \r\n" );
document.write( "    1 + cos(2pi/7)+cos(4pi/7)+cos(6pi/7)+cos(8pi/7)+cos(10pi/7)+cos(12pi/7) = 0.     (1)\r\n" );
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document.write( "    Notice that \"1\", the first addend, is x-coordinate of the point A.\r\n" );
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document.write( "3.  From symmetry, it is also clear that the x-coordinates of the pairs of the points B and G; C and F; D and E are the same:\r\n" );
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document.write( "    cos(2pi/7) = cos(12pi/7);  cos(4pi/7) = cos(10pi/7);  cos(6pi/7) = cos(8pi/7).\r\n" );
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document.write( "    Therefore, we can re-write the sum (1) in the form\r\n" );
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document.write( "    2*(cos(2pi/7)+cos(4pi/7)+cos(6pi/7)) = -1.\r\n" );
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document.write( "4.  Now all that we need to do is to divide both sides of (2) by 2, and we will get \r\n" );
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document.write( "       cos(2pi/7)+cos(4pi/7)+cos(6pi/7) = ,\r\n" );
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document.write( "    which is the same as \r\n" );
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document.write( "       cos(2pi/7)+cos(4pi/7)+cos(8pi/7) = , \r\n" );
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document.write( "    as we noted above.\r\n" );
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document.write( "5.  Proved.\r\n" );
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