document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138824>Question 138824</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%2F2%29%29%5E2-%28sin%28x%29%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28sin%28x%2F2%29%29%5E2-%28sin%28x%29%29%5E2=0\"   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 #101281 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=101281\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(\frac{\theta}{2}\right)-\sin^2\left(\theta\right)=0 \"> Start with the given equation\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=\frac{\theta}{2}\">. So this means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2u=2*\frac{\theta}{2}=\theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)-\sin^2\left(2u\right)=0  \"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\theta}{2}\"> with u and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta\"> with 2u\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)-\left(\sin\left(2u\right)\right)^2=0  \"> Rewrite <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(2u\right)\"> as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\sin\left(2u\right)\right)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)-\left(2\sin\left(u\right)\cos\left(u\right)\right)^2=0 \"> Replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(2u\right)\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\sin\left(u\right)\cos\left(u\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)-4\sin^2\left(u\right)\cos^2\left(u\right)=0 \"> Square <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\sin\left(u\right)\cos\left(u\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)\left(1-4\cos^2\left(u\right)\right)=0 \"> Factor out <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)\">\r<BR>\n" );
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document.write( "Now use the zero product property:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)=0\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1-4\cos^2\left(u\right)=0 \">\r<BR>\n" );
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document.write( "Now let's solve <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)=0\">: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin^2\left(u\right)=0 \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin\left(u\right)=0 \"> Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=0\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=\pi \"> Take the arcsine of both sides\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=\frac{\theta}{2}\">, this means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=0\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=2\pi \">\r<BR>\n" );
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document.write( "However, since  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2\pi \"> is not in the interval [0,2<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \pi \">), it is not a solution.\r<BR>\n" );
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document.write( "So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=0\">\r<BR>\n" );
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document.write( "--------------------------------------------------------------------\r<BR>\n" );
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document.write( "Now let's solve <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1-4\cos^2\left(u\right)=0 \">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1-4\cos^2\left(u\right)=0 \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  -4\cos^2\left(u\right)=-1 \"> Subtract 1 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  \cos^2\left(u\right)=\frac{1}{4} \"> Divide both sides by -4\r<BR>\n" );
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document.write( "Take the square root of both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  \cos\left(u\right)=\frac{1}{2}\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(u\right)=-\frac{1}{2} \"> \r<BR>\n" );
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document.write( "So let's solve the first part <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(u\right)=\frac{1}{2} \">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(u\right)=\frac{1}{2} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=\frac{\pi}{3}\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=-\frac{\pi}{3} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\frac{2\pi}{3}\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=-\frac{2\pi}{3} \">\r<BR>\n" );
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document.write( "However since our interval is positive, the negative answer is not in the interval.\r<BR>\n" );
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document.write( "So another part of the solution is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\frac{2\pi}{3}\">\r<BR>\n" );
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document.write( "---------------------\r<BR>\n" );
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document.write( "Now let's solve the second part <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(u\right)=-\frac{1}{2} \">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(u\right)=-\frac{1}{2} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=\frac{2\pi}{3}\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u=-\frac{2\pi}{3} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\frac{4\pi}{3}\">  ...or...  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=-\frac{4\pi}{3} \">\r<BR>\n" );
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document.write( "So another part of the solution is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\frac{4\pi}{3}\">\r<BR>\n" );
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document.write( "===============================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So all together our solutions are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=0,\theta=\frac{2\pi}{3}, \hspace {12} \textrm{or} \hspace {12} \theta=\frac{4\pi}{3}\"> </SPAN>\n" );
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