document.write( "Question 1129915: Find the sum of 1sin(1 degree)+2sin(2 degrees)+3sin(3 degrees)+4sin(4 degrees)+...+178sin(178 degrees)+179sin(179 degrees)+180sin(180 degrees). \n" ); document.write( "
Algebra.Com's Answer #746567 by ikleyn(52754)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let  R = 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°).\r\n" );
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document.write( "It is even more convenient for me to write\r\n" );
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document.write( "    R = 0*sin(0°) + 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)  by adding  0*sin(0°) = 0  as the first term.\r\n" );
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document.write( "Then I can write\r\n" );
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document.write( "    R =   0*sin(0°)   + 1*sin(1°)     + 2*sin(2°)    + 3*sin(3°)     + . . . + 180*sin(180°)     (1)\r\n" );
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document.write( "    R = 180*sin(180°) + 179*sin(179°) + 178*sin178°) + 177*sin(177°) + . . . + 0*sin(0°).        (2)\r\n" );
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document.write( "Note that sin(0°) = sin(180°);  sin(1°) = sin(179°);  sin(2°) = sin(178°);  sin(3°) = sin(177°);  and so on.   \r\n" );
document.write( "Therefore, adding  (1)  and  (2)\r\n" );
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document.write( "   2R = 180*(sin(0°) + sin(1°) + sin(2°) + sin(3°) + . . . + sin(90°)).\r\n" );
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document.write( "So, our sum R is the same as\r\n" );
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document.write( "     R = 90*(sin(0°) + sin(1°) + sin(2°) + sin(3°) + . . . + sin(90°)).\r\n" );
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document.write( "Now, let z = cos(1°) + i*sin(1°)  is this complex number.\r\n" );
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document.write( "Then, according to the de Moivre's formula\r\n" );
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document.write( "         \"z%5E2\" = cos(2°) + i*sin(2°)\r\n" );
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document.write( "         \"z%5E3\" = cos(3°) + i*sin(3°),\r\n" );
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document.write( "              .  .  .  .  .  .\r\n" );
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document.write( "          \"z%5E90\" = cos(90° + i*sin(90°),\r\n" );
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document.write( "and the sum R  is 90 times the imaginary part  of the sum of the geometric progression\r\n" );
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document.write( "          Q = 1 + \"z\" + \"z%5E2\" + \"z%5E3\" + \"z%5E4\" + . . . + \"z%5E90\".\r\n" );
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document.write( "The sum of the geometric progression is  \r\n" );
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document.write( "    \"%28z%5E91-1%29%2F%28z-1%29\" =  = now I will work to extract the imaginary part =  = \r\n" );
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document.write( "  =  . \r\n" );
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document.write( "So, the imaginary part has the DENOMINATOR  (cos(1°)-1)^2+sin^2(1°) = 2-2*cos(1°) = 2*(1-cos(1°)).\r\n" );
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document.write( "The imaginary part has the NUMERATOR  sin(91°)*(cos(1°)-1) - (cos(91°)-1)*sin(1°) = sin(91°)*cos(1°) - cos(91°)*sin(1°) - sin(91°) + sin(1°) = \r\n" );
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document.write( "                                    = sin(90°) - sin(91°) + sin(1°) = 1 - cos(1°) + sin(1°).\r\n" );
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document.write( "Therefore, the answer is:  \r\n" );
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document.write( "    the sum  R = 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°) = \"90%2A%28%281+-+cos%281%5Eo%29+%2B+sin%281%5Eo%29%29%2F%282%2A%281-cos%281%5Eo%29%29%29%29\".\r\n" );
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document.write( "You can transform it further as you want and/or as you need.\r\n" );
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document.write( "You can even get the numerical value, using your calculator.\r\n" );
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