document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152146>Question 1152146</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+3x+=+cos+x ALIGN=MIDDLE  ALT=\"cos+3x+=+cos+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval [0,2pi) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774084 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=774084\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Find all solutions of the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%283x%29=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%283x%29=cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   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=cos%283x%29=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%283x%29=cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >......write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%283x%29 ALIGN=MIDDLE  ALT=\"cos%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%2Bx%29 ALIGN=MIDDLE  ALT=\"cos%282x%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and use identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha+%2B+beta%29+=+cos%28alpha%29+cos%28beta%29+-+sin%28alpha%29+sin%28beta%29 ALIGN=MIDDLE  ALT=\"cos%28alpha+%2B+beta%29+=+cos%28alpha%29+cos%28beta%29+-+sin%28alpha%29+sin%28beta%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=cos%282x%2Bx%29=+cos%282x%29+cos%28x%29++-++sin%282x%29+sin%28x%29 ALIGN=MIDDLE  ALT=\"cos%282x%2Bx%29=+cos%282x%29+cos%28x%29++-++sin%282x%29+sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29 ALIGN=MIDDLE  ALT=\"cos%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2Bx%29 ALIGN=MIDDLE  ALT=\"cos%28x%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and use identity above\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2Bx%29=+cos%28x%29+cos%28x%29++-++sin%28x%29+sin%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%2Bx%29=+cos%28x%29+cos%28x%29++-++sin%28x%29+sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282x%29=sin%28x%2Bx%29=sin%28x%29+cos%28x%29+%2B+cos%28x%29+sin%28x%29 ALIGN=MIDDLE  ALT=\"sin%282x%29=sin%28x%2Bx%29=sin%28x%29+cos%28x%29+%2B+cos%28x%29+sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%283x%29=%28cos%28x%29+cos%28x%29+-sin%28x%29+sin%28x%29%29+cos%28x%29+-+%28sin%28x%29+cos%28x%29+%2B+cos%28x%29+sin%28x%29%29+sin%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "substitute in given equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%283x%29=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%283x%29=cos%28x%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=%28cos%28x%29+cos%28x%29+-sin%28x%29+sin%28x%29%29+cos%28x%29+-+%28sin%28x%29+cos%28x%29+%2B+cos%28x%29+sin%28x%29%29+sin%28x%29=cos%28x%29 ALIGN=MIDDLE   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=%28cos%5E2%28x%29++-sin%5E2%28x%29+%29+cos%28x%29+-+%28sin%5E2%28x%29+cos%28x%29+%2B+cos%28x%29+sin%5E2%28x%29%29+=cos%28x%29 ALIGN=MIDDLE   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=%28cos%5E2%28x%29++-sin%5E2%28x%29+%29+cos%28x%29+-+%28sin%5E2%28x%29++%2B+sin%5E2%28x%29%29+cos%28x%29+=cos%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >....both sides divide by  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%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=%28cos%5E2%28x%29++-sin%5E2%28x%29+%29++-+%282sin%5E2%28x%29+%29++=1 ALIGN=MIDDLE  ALT=\"%28cos%5E2%28x%29++-sin%5E2%28x%29+%29++-+%282sin%5E2%28x%29+%29++=1\"   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=cos%5E2%28x%29++-sin%5E2%28x%29+++-+2sin%5E2%28x%29+++=1 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29++-sin%5E2%28x%29+++-+2sin%5E2%28x%29+++=1\"   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=cos%5E2%28x%29++-3sin%5E2%28x%29+++=1 ALIGN=MIDDLE  ALT=\"cos%5E2%28x%29++-3sin%5E2%28x%29+++=1\"   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=++1-cos%5E2%28x%29%2B3sin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"++1-cos%5E2%28x%29%2B3sin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........use the identity: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-cos+%5E2%28x%29=sin+%5E2%28x%29 ALIGN=MIDDLE  ALT=\"1-cos+%5E2%28x%29=sin+%5E2%28x%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=sin+%5E2%28x%29%2B3sin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"sin+%5E2%28x%29%2B3sin%5E2%28x%29=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=4sin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"4sin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=> solutions:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...........apply rule: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5En=0 ALIGN=MIDDLE  ALT=\"x%5En=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><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;\" >\r<BR>\n" );
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document.write( " general solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B2pi%2An ALIGN=MIDDLE  ALT=\"x=0%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=2pi%2An ALIGN=MIDDLE  ALT=\"+x=2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2B2pi%2An ALIGN=MIDDLE  ALT=\"x=pi%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions for the range : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C2pi+ ALIGN=MIDDLE  ALT=\"0%3C=x%3C2pi+\"   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=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "combine solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F2 ALIGN=MIDDLE  ALT=\"x=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3pi%2F2 ALIGN=MIDDLE  ALT=\"x=3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );