document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=596831>Question 596831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+cos%5E2%28x%29+-+5cos%28x%29+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"2+cos%5E2%28x%29+-+5cos%28x%29+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all principal values of x.\r<BR>\n" );
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document.write( "I am not sure where nor how to proceed with problem. Am I supposed to use a Pythagorean identity, and if so, where do I go from there. Any help you can give me would be so greatly appreciated. I am struggling through my precalculus book, and I did not do well in Trig or Geometry and passes Algebra I and II with a mid C. Please help me understand this problem. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #377910 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=377910\">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( "First of all, I fixed your equation.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(x^2)\"> means the cosine of the angle <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2\">, whereas  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^2(x)\"> is the squared value of the cosine function at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">.  Very different things and they didn't give you\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\cos(x^2)\ -\ 5\cos(x)\ +\ 2\ =\ 0\">\r<BR>\n" );
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document.write( "in a Precalculus book. \r<BR>\n" );
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document.write( "The easiest way to see what is going on in this equation is to use a simple substitution.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u\ =\ \cos(x)\"> and substitute.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u^2\ -\ 5u\ +\ 2\ =\ 0\">\r<BR>\n" );
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document.write( "Factor the quadratic in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (2u\ -\ 1)(u\ -\ 2)\ =\ 0\">\r<BR>\n" );
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document.write( "Apply the Zero Product Rule:\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( "or\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( "Now recall the RANGE of the cosine function, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -1\ \leq\ \cos(\varphi)\ \leq\ 1\">.  That means that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u\ =\ 2\"> is an extraneous root and can be discarded.\r<BR>\n" );
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document.write( "Hence, substituting back\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos(x)\ =\ \frac{1}{2}\">\r<BR>\n" );
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document.write( "The principal values of the argument for cosine are in the interval\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0\ \leq\ \cos(x)\ \leq\ \pi\">\r<BR>\n" );
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document.write( "Find the two angles in the unit circle where the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-coordinate is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{2}\"> and select the one that is in the principal value interval.\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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{\pi}{3}\">\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( "My calculator said it, I believe it, that settles it<BR>\n" );
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