document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277195>Question 277195</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve for theta if 2 tan^2 theta + sec theta-4=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #201861 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=201861\">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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\tan^2\theta\ +\ sec\theta\ -\ 4\ =\ 0\">\r<BR>\n" );
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document.write( "Recall that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \ \ \tan\phi\ =\ \frac{\sin\phi}{\cos\phi}\ \ \ \"> and that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \ \ \sec\phi\ =\ \frac{1}{\cos\phi}\ \ \ \">,  hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{2\sin^2\theta}{\cos^2\theta}\ +\ \frac{1}{\cos\theta}\ -\ 4\ =\ 0\">\r<BR>\n" );
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document.write( "LCD is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large cos^2\theta\"> so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{2\sin^2\theta}{\cos^2\theta}\ +\ \frac{\cos\theta}{\cos^2\theta}\ -\ \frac{4\cos^2\theta}{\cos^2\theta}\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\sin^2\theta\ +\ \cos\theta\ -\ 4\cos^2\theta\ =\ 0\">\r<BR>\n" );
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document.write( "Now recall that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large cos^2\phi\ +\ \sin^2\phi\ =\ 1\"> so that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin^2\phi\ =\ 1\ -\ cos^2\phi\"> and therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(1\ -\ cos^2\theta\right)\ +\ \cos\theta\ -\ 4\cos^2\theta\ =\ 0\">\r<BR>\n" );
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document.write( "and finally:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 6\cos^2\theta\ -\ \cos\theta\ -\ 2\ = 0\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \cos\theta\">, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 6x^2\ -\ x\ -\ 2\ = 0\">\r<BR>\n" );
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document.write( "Which factors to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(3x\ -\ 2\right)\left(2x\ +\ 1\right)\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{2}{3}\">\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{1}{2}\">\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\theta\ =\ \frac{2}{3}\">\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 \ \ \ \ \ \ \ \ \ \ \cos\theta\ =\ -\frac{1}{2}\">\r<BR>\n" );
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document.write( "And\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \theta\ =\ \cos^{-1}\left(\frac{2}{3}\right)\ \approx\ 0.841\"> radians or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 5.442\"> radians\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 \ \ \ \ \ \ \ \ \ \ \theta\ =\ \cos^{-1}\left(-\frac{1}{2}\right)\ =\ \frac{2\pi}{3}\"> radians or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{4\pi}{3}\"> radians.\r<BR>\n" );
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document.write( "Though you did not specify, I presumed you wanted the solutions on the interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ 0\ \leq\ \theta\ \leq\ 2\pi\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png\">\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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