document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=324020>Question 324020</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions for the equation. Record your answers in exact form where possible.<BR>\n" );
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document.write( "2 cos ^2 x = -3sin x\r<BR>\n" );
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document.write( "Thanks for your help, any help is appreciated  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #231902 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=231902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "First thing to do is remember that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos^2(x)\ +\ \sin^2(x)\ =\ 1\">\r<BR>\n" );
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document.write( "Which means that you can re-cast your equation thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2(1\ -\ \sin^2(x))\ =\ -3\sin(x)\">\r<BR>\n" );
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document.write( "and then a little re-arranging:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\sin^2(x)\ -\ 3\sin(x)\ -\ 2\ =\ 0\">\r<BR>\n" );
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document.write( "Next, let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ \sin(x)\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u^2\ -\ 3u\ -\ 2\ =\ 0\">\r<BR>\n" );
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document.write( "which factors rather tidily:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (u\ -\ 2)(2u\ +\ 1)\ =\ 0\">\r<BR>\n" );
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document.write( "Hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ u\ =\ 2\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ u\ =\ -\frac{1}{2}\">\r<BR>\n" );
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document.write( "But recall that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ \sin(x)\">.  Since the range of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin(\varphi)\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left[-1\ \leq\ \sin(\varphi)\ \leq\ 1\right]\">, we must exclude <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ 2\">.  Therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ u\ \sin(x)\ =\ -\frac{1}{2}\">\r<BR>\n" );
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document.write( "Finally,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \sin^{-1}\left(-\frac{1}{2}\right)\">\r<BR>\n" );
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document.write( "Looking at the unit circle and remembering that sin is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of a point on the unit circle,\r<BR>\n" );
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document.write( "(see http://en.wikipedia.org/wiki/Unit_circle)\r<BR>\n" );
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document.write( "You see that there are two possibilities, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{7\pi}{6}\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{11\pi}{6}\">.  However, the instructions for the problem say to find <i><b>all</b></i> solutions and does not give a range, hence we need to allow for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\pi\"> periodicity of the sine function.  Therefore the final answer is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{7\pi}{6}\ +\ 2k\pi\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{11\pi}{6}\ +\ 2k\pi\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ \in\ \mathbb{Z}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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