document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=174444>Question 174444</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 60 find all the solution in the interval 0,2pi<BR>\n" );
document.write( "  sin(x+pi/6)-sin(x-pi/6)=1/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #129436 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=129436\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%2Bpi%2F6%29-sin%28x-pi%2F6%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29%29-sin%28x-pi%2F6%29=1%2F2 ALIGN=MIDDLE  ALT=\"%28sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29%29-sin%28x-pi%2F6%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2Bpi%2F6%29 ALIGN=MIDDLE  ALT=\"sin%28x%2Bpi%2F6%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=%28sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29%29-%28sin%28x%29cos%28pi%2F6%29-cos%28x%29sin%28pi%2F6%29%29=1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Expand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x-pi%2F6%29 ALIGN=MIDDLE  ALT=\"sin%28x-pi%2F6%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%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29-sin%28x%29cos%28pi%2F6%29%2Bcos%28x%29sin%28pi%2F6%29=1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%28sqrt%283%29%2F2%29%2Bcos%28x%29%281%2F2%29-sin%28x%29%28sqrt%283%29%2F2%29%2Bcos%28x%29%281%2F2%29=1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F6%29 ALIGN=MIDDLE  ALT=\"cos%28pi%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F6%29 ALIGN=MIDDLE  ALT=\"sin%28pi%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   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=%28sqrt%283%29%2F2%29sin%28x%29%2B%281%2F2%29cos%28x%29-%28sqrt%283%29%2F2%29sin%28x%29%2B%281%2F2%29cos%28x%29=1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28x%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=arccos%281%2F2%29 ALIGN=MIDDLE  ALT=\"cos%28x%29=arccos%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the arccosine of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F3 ALIGN=MIDDLE  ALT=\"x=pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-pi%2F3 ALIGN=MIDDLE  ALT=\"x=-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the arccosine of 1/2 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F3 ALIGN=MIDDLE  ALT=\"-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-pi%2F3 ALIGN=MIDDLE  ALT=\"x=-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT in the range [0,2pi), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F3 ALIGN=MIDDLE  ALT=\"-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT a solution\r<BR>\n" );
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document.write( "So the first solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F3 ALIGN=MIDDLE  ALT=\"x=pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2Bpi%2F3=6pi%2F3%2Bpi%2F3=%287pi%29%2F3 ALIGN=MIDDLE  ALT=\"2pi%2Bpi%2F3=6pi%2F3%2Bpi%2F3=%287pi%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Take note how this value is outside the interval [0,2pi). So this value is NOT a solution\r<BR>\n" );
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document.write( "Now add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F3 ALIGN=MIDDLE  ALT=\"-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi-pi%2F3=6pi%2F3-pi%2F3=%285pi%29%2F3 ALIGN=MIDDLE  ALT=\"2pi-pi%2F3=6pi%2F3-pi%2F3=%285pi%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this value is inside the interval [0,2pi), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285pi%29%2F3 ALIGN=MIDDLE  ALT=\"%285pi%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F3 ALIGN=MIDDLE  ALT=\"x=pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285pi%29%2F3 ALIGN=MIDDLE  ALT=\"%285pi%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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