document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=797441>Question 797441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solving Trig Equations<BR>\n" );
document.write( "What do I do if my trig equation does not equal a reference triangle?<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "sec 2 theta = 1<BR>\n" );
document.write( "I know that sec=Hypotenuse/adjacent, but how do I graph a triangle with a hypotenuse of 1 and an adjacent angle of 1? I am familiar with graphing points on a plane (ex: cos theta = 1, graph at point (1,0)) but 1) I am using secant, the opposite of cosine, and 2) I am working with secant 2 theta, not just 1 theta. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #481807 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=481807\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Cosine is NOT the opposite of secant.  That would mean that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\varphi\ =\ -\sec\varphi\">  FALSE!  Cosine is the reciprocal of secant.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\varphi\ =\ \frac{1}{\sec\varphi}\">.\r<BR>\n" );
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document.write( "So:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sec 2\theta\ =\ \frac{1}{\cos 2\theta}\">.  But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sec 2\theta\ =\ 1,\ \ \"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{1}{\cos 2\theta}\ =\ 1\ \Rightarrow\ \cos 2\theta\ =\ 1\">.\r<BR>\n" );
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document.write( "Now you need the double angle formula for cosine: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos 2\varphi\ =\ \cos^2\varphi\ -\ \sin^2\varphi\">.  So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\theta\ -\ \sin^2\theta\ =\ 1\">\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin^2\varphi\ =\ 1\ -\ \cos^2\varphi\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\theta\ -\ \left(1\ -\ \cos^2\theta\right)\ =\ 1\">\r<BR>\n" );
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document.write( "A little algebra music, Sammy\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\cos^2\theta\ =\ 2\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos^2\theta\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\theta\ =\ \pm1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \theta\ =\ \arccos(1)\ =\ 0\ +\ 2k\pi\ \forall k\ \in\ \mathbb{Z}\"> \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\ =\ \arccos(-1)\ =\ \pi\ +\ 2k\pi\ \forall 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" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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