document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103012>Question 1103012</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a. The equation r2 = 4cos2&#952; is of the form r^2 = a^2*cos2&#952;, so the graph is a lemniscate. What is the length of each loop?\r<BR>\n" );
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document.write( "b. At what values of &#952; do the endpoints of each loop occur? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717736 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717736\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For a lemniscate of the form,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=a%5E2%2Acos%282%2Atheta%29 ALIGN=MIDDLE  ALT=\"r%5E2=a%5E2%2Acos%282%2Atheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the arc length is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28a%2Fsqrt%282pi%29%29%2AGAMMA%281%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"S=%28a%2Fsqrt%282pi%29%29%2AGAMMA%281%2F4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GAMMA ALIGN=MIDDLE  ALT=\"GAMMA\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the gamma function.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=GAMMA%281%2F2%29=sqrt%28pi%29 ALIGN=MIDDLE  ALT=\"GAMMA%281%2F2%29=sqrt%28pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%282%2Fsqrt%282pi%29%29pi ALIGN=MIDDLE  ALT=\"s=%282%2Fsqrt%282pi%29%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=sqrt%282pi%29 ALIGN=MIDDLE  ALT=\"s=sqrt%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
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document.write( "The maximum absolute value of the cosine is equal to 1 and occurs when the argument <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Atheta=0 ALIGN=MIDDLE  ALT=\"2%2Atheta=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Atheta=pi ALIGN=MIDDLE  ALT=\"2%2Atheta=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So then when,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=0 ALIGN=MIDDLE  ALT=\"theta=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%281%2F2%29pi ALIGN=MIDDLE  ALT=\"theta=%281%2F2%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi ALIGN=MIDDLE  ALT=\"theta=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%283%2F2%29pi ALIGN=MIDDLE  ALT=\"theta=%283%2F2%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and equals zero when,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%281%2F4%29pi ALIGN=MIDDLE  ALT=\"theta=%281%2F4%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%283%2F4%29pi ALIGN=MIDDLE  ALT=\"theta=%283%2F4%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%285%2F4%29pi ALIGN=MIDDLE  ALT=\"theta=%285%2F4%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=%287%2F4%29pi ALIGN=MIDDLE  ALT=\"theta=%287%2F4%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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