document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203823>Question 1203823</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all exact solutions on [0, 2𝜋).  (Enter your answers as a comma-separated list.)<BR>\n" );
document.write( "2 cos^2(t) − cos(t) = 1<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 #839669 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let w = cos(t)\r<BR>\n" );
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document.write( "The original equation <BR>\n" );
document.write( "2cos^2(t) - cos(t) = 1<BR>\n" );
document.write( "turns into<BR>\n" );
document.write( "2w^2 - w = 1<BR>\n" );
document.write( "and rearranges into<BR>\n" );
document.write( "2w^2 - w - 1 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula, or factoring, to find the roots are:<BR>\n" );
document.write( "w = -1/2 or w = 1\r<BR>\n" );
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document.write( "This leads us back to<BR>\n" );
document.write( "cos(t) = -1/2 or cos(t) = 1\r<BR>\n" );
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document.write( "Then use the unit circle to find that cos(t) = -1/2 has the solutions: <font color=red>t = 2pi/3 and t = 4pi/3</font><BR>\n" );
document.write( "In other words,<BR>\n" );
document.write( "cos(2pi/3) = -1/2 and cos(4pi/3) = -1/2 when your calculator is set to radian mode.\r<BR>\n" );
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document.write( "Use the unit circle to find the solutions to cos(t) = 1 are: <font color=red>t = 0</font> and t = 2pi<BR>\n" );
document.write( "However, we exclude 2pi due to the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+t+%3C+2pi ALIGN=MIDDLE  ALT=\"0+%3C=+t+%3C+2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > which condenses to the interval notation [0, 2pi)\r<BR>\n" );
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document.write( "You can use a graphing calculator such as a TI83/TI84, Desmos, or GeoGebra to confirm these answers.<BR>\n" );
document.write( "The function to plot would be f(x) = 2*(cos(x))^2 - cos(x) - 1<BR>\n" );
document.write( "The goal is to look for the x intercepts on the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C+2pi ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C+2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that<BR>\n" );
document.write( "2pi/3 = 2.094395 approximately<BR>\n" );
document.write( "4pi/3 = 4.188790 approximately\r<BR>\n" );
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document.write( "Answers:  <font color=red size=4>0, 2pi/3, 4pi/3</font><BR>\n" );
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