document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=457926>Question 457926</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28cos%28pi%28k-5%29%2F20%29%2C+k+=+0%2C+20%29%2Fsum%28sin%28%28k%2Api%29%2F20%29%2C+k+=+0%2C+20%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #314189 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=314189\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We can use the sum/difference formulas for cosine to simplify the expression to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{ \sum_{k=0}^{20} \cos{\frac{k\pi}{20}} \cos{\frac{\pi}{4}} + \sin{\frac{k\pi}{20}} \sin{\frac{\pi}{4}}  } {\sum_{k=0}^{20} \sin(\frac{k\pi}{20})} = \sqrt{2} + \sqrt{2}\frac{ \sum_{k=0}^{20} \cos{\frac{k\pi}{20}}}{\sum_{k=0}^{20} \sin{\frac{k\pi}{20}}\">\r<BR>\n" );
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document.write( "Note that the expression <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sum_{k=0}^{20} \cos{\frac{k\pi}{20}\"> is equal to zero, because if you were to draw all the angles on a unit circle, you would see that there are corresponding pairs of angles that add up to zero, since cos x = -cos (pi - x). The denominator of the fraction is some positive number because the sine of each angle is either positive or zero. Therefore the entire expression is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\sqrt{2}\">. </SPAN>\n" );
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