document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=924330>Question 924330</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please solve the following equation. <BR>\n" );
document.write( "(Enter your answers as a comma-separated list. Let k be any integer. Enter DNE is there is no solution.)<BR>\n" );
document.write( "2 cos 2&#952; &#8722; 1 = 0<BR>\n" );
document.write( "(a) Find all solutions of the equation.<BR>\n" );
document.write( "&#952; =   \r<BR>\n" );
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document.write( "(b) Find the solutions in the interval [0, 2&#960;).<BR>\n" );
document.write( "&#952; =   \r<BR>\n" );
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document.write( "Thanks  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #560765 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=560765\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Part a)\r<BR>\n" );
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document.write( "Solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta=+pi%2F3%2B2pi%2Ak ALIGN=MIDDLE  ALT=\"2theta=+pi%2F3%2B2pi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta=+-pi%2F3%2B2pi%2Ak ALIGN=MIDDLE  ALT=\"2theta=+-pi%2F3%2B2pi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k is any integer.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+%281%2F2%29%2A%28pi%2F3%2B2pi%2Ak%29 ALIGN=MIDDLE  ALT=\"theta=+%281%2F2%29%2A%28pi%2F3%2B2pi%2Ak%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+%281%2F2%29%2A%28-pi%2F3%2B2pi%2Ak%29 ALIGN=MIDDLE  ALT=\"theta=+%281%2F2%29%2A%28-pi%2F3%2B2pi%2Ak%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+-pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+-pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The general solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+-pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+-pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k is any integer.\r<BR>\n" );
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document.write( "You can condense that into one equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+%22%22%2B-pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+%22%22%2B-pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > (take note of the plus/minus)\r<BR>\n" );
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document.write( "======================================================================================================================\r<BR>\n" );
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document.write( "Part b)\r<BR>\n" );
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document.write( "We'll find the first set of solutions.\r<BR>\n" );
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document.write( "Plug in integer values of k into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > to generate solutions. Make sure you are between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2pi+=+6.28 ALIGN=MIDDLE  ALT=\"x+=+2pi+=+6.28\"   DISABLED_style= \"max-width:90%; height:auto;\" > roughly.\r<BR>\n" );
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document.write( "Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=++0 ALIGN=MIDDLE  ALT=\"k+=++0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+0.5235987755983 ALIGN=MIDDLE  ALT=\"theta=+0.5235987755983\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is in the interval [0,2pi), so we keep it.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+3.6651914291881 ALIGN=MIDDLE  ALT=\"theta=+3.6651914291881\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is in the interval [0,2pi), so we keep it.\r<BR>\n" );
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document.write( "Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=++2 ALIGN=MIDDLE  ALT=\"k+=++2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we have gone over 6.28, so we stop here and we don't include this solution because it is not in the interval [0,2pi)\r<BR>\n" );
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document.write( "Now we move onto the second part.\r<BR>\n" );
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document.write( "Plug in integer values of k into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+-pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+-pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" > to generate solutions. Make sure you are between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2pi+=+6.28 ALIGN=MIDDLE  ALT=\"x+=+2pi+=+6.28\"   DISABLED_style= \"max-width:90%; height:auto;\" > roughly.\r<BR>\n" );
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document.write( "That is less than x = 0, so we toss out this solution.\r<BR>\n" );
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document.write( "This is in the interval [0,2pi), so we keep it.\r<BR>\n" );
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document.write( "This is in the interval [0,2pi), so we keep it.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+-pi%2F6%2Bpi%2Ak ALIGN=MIDDLE  ALT=\"theta=+-pi%2F6%2Bpi%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+-pi%2F6%2Bpi%2A3 ALIGN=MIDDLE  ALT=\"theta=+-pi%2F6%2Bpi%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=+8.9011791851711 ALIGN=MIDDLE  ALT=\"theta=+8.9011791851711\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we have gone over 6.28, so we stop here and we don't include this solution because it is not in the interval [0,2pi)\r<BR>\n" );
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document.write( "---------------------------------\r<BR>\n" );
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document.write( "The exact solutions to part b) are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F6 ALIGN=MIDDLE  ALT=\"x+=+pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+7pi%2F6 ALIGN=MIDDLE  ALT=\"x+=+7pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5pi%2F6 ALIGN=MIDDLE  ALT=\"x+=+5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+11pi%2F6 ALIGN=MIDDLE  ALT=\"x+=+11pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. They are exact solutions in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Let me know if that helps or not. Thanks.\r<BR>\n" );
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document.write( "If you need more help, feel free to email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "My Website: <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">http://www.freewebs.com/jimthompson5910/home.html</a><BR>\n" );
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