document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=611643>Question 611643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following equation for the principal values:<BR>\n" );
document.write( "2cos^3 x+2cos^2 x-cosx-1=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #385129 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=385129\">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( "I'm going to presume you mean \"for the principal values of the inverse cosine\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\cos^3(x)\ +\ 2\cos^2(x)\ -\ \cos(x)\ -\ 1\ =\ 0\"> \r<BR>\n" );
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document.write( "This is much easier to see if you use substitution.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u\ =\ \cos(x)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u^3\ +\ 2u^2\ -\ u\ -\ 1\ =\ 0\">  \r<BR>\n" );
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document.write( "Rearrange terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2u^3\ -\ u\ +\ 2u^2\ -\ 1\ =\ 0\"> \r<BR>\n" );
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document.write( "Factor a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE u\"> from the first two terms and a 1 from the second pair of terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ u\left(2u^2\ -\ 1\right)\ +\ \left(2u^2\ -\ 1\right)\ =\ 0\">\r<BR>\n" );
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document.write( "Then factor out <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2u^2\ -\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(u\ +\ 1\right)\left(2u^2\ -\ 1\right)\ =\ 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 \ \ \ \ \ \ \ \ \ \ \cos(x)\ =\ -1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos(x)\ =\ \pm\frac{\sqrt{2}}{2}\">\r<BR>\n" );
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document.write( "The principal values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^{-1}(z)\"> are in the closed interval <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\frac{\pi}{2}\ \leq\ z\ \leq\ \frac{\pi}{2}\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(x)\ \geq\ 0\ \forall\,x\,\in\ -\frac{\pi}{2}\ \leq\ z\ \leq\ \frac{\pi}{2}\">\r<BR>\n" );
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document.write( "Discard both negative roots.  Use the unit circle to find the two values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> in the principal interval that satisfy the equation.  Hint:  cosine  is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> coordinate of a point on the unit circle.  Anothe hint: Angle values in the 4th quadrant from the point of view of the pricipal interval are negative values.\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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