document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=570994>Question 570994</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a genius figured out that\r<BR>\n" );
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document.write( "sin(7pi/12)= -((sqrt(2)+sqrt(60)/4) \r<BR>\n" );
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document.write( "Find each of the following exactly (show steps)\r<BR>\n" );
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document.write( "a). sin (-(7pi/12))\r<BR>\n" );
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document.write( "b.) sin (-(5pi/12))\r<BR>\n" );
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document.write( "c. cos(pi/12) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #368128 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=368128\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It does not take a genius to find the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287%2Api%2F12%29 ALIGN=MIDDLE  ALT=\"sin%287%2Api%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but having those trigonometric identity formulas handy helps.<BR>\n" );
document.write( "The expression you posted for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287%2Api%2F12%29 ALIGN=MIDDLE  ALT=\"sin%287%2Api%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is wrong. Either someone made a typo somewhere, or the person who wrote the problem is trying to confuse us all.<BR>\n" );
document.write( "Since the expression given for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287%2Api%2F12%29 ALIGN=MIDDLE  ALT=\"sin%287%2Api%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > looked fishy to me, I went looking for the trigonometric identity formulas to find the correct exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287%2Api%2F12%29 ALIGN=MIDDLE  ALT=\"sin%287%2Api%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%287%2Api%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%287%2Api%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "HOW I CALCULATED THAT (just in case you care)<BR>\n" );
document.write( "I found the trigonometric identity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%2BB%29=sin%28A%29cos%28B%29%2Bcos%28A%29sin%28B%29 ALIGN=MIDDLE  ALT=\"sin%28A%2BB%29=sin%28A%29cos%28B%29%2Bcos%28A%29sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and that was useful, because I know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%2B1%2F3=3%2F12%2B4%2F12=7%2F12 ALIGN=MIDDLE  ALT=\"1%2F4%2B1%2F3=3%2F12%2B4%2F12=7%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4%2Bpi%2F3=7pi%2F12 ALIGN=MIDDLE  ALT=\"pi%2F4%2Bpi%2F3=7pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and everybody knows that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F4%29=cos%28pi%2F4%29=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28pi%2F4%29=cos%28pi%2F4%29=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F3%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28pi%2F3%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F3%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28pi%2F3%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287%2Api%2F12%29=sin%28pi%2F4%2Bpi%2F3%29=sin%28pi%2F4%29cos%28pi%2F3%29%2Bcos%28pi%2F4%29sin%28pi%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >=<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29%2F2%29%281%2F2%29%2B%28sqrt%282%29%2F2%29%28sqrt%283%29%2F2%29=sqrt%282%29%2F4%2Bsqrt%282%29sqrt%283%29%2F4+=+sqrt%282%29%2F4%2Bsqrt%286%29%2F4=%28sqrt%282%29%2Bsqrt%286%29%29%2F4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "BACK TO THE PROBLEM<BR>\n" );
document.write( "I am going to use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%287%2Api%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%287%2Api%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "However, it turns out that all the answers are either that expression, or (-1) times that, so if you were meant to use the fishy expression, you'll easily figure out the intended answers<BR>\n" );
document.write( "a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-anything%29=-sin%28anything%29 ALIGN=MIDDLE  ALT=\"sin%28-anything%29=-sin%28anything%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-7pi%2F12%29=-sin%287pi%2F12%29=highlight%28-%28%28sqrt%282%29%2Bsqrt%286%29%29%2F4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F12%2B7pi%2F12=12pi%2F12=pi ALIGN=MIDDLE  ALT=\"5pi%2F12%2B7pi%2F12=12pi%2F12=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F12 ALIGN=MIDDLE  ALT=\"5pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7pi%2F12 ALIGN=MIDDLE  ALT=\"7pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > are supplementary angles. They add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "And we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29sin%28pi-A%29 ALIGN=MIDDLE  ALT=\"sin%28A%29sin%28pi-A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%285pi%2F12%29=sin%287pi%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"sin%285pi%2F12%29=sin%287pi%2F12%29=%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-5pi%2F12%29=-sin%28-pi%2F12%29=highlight%28-%28%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F12=7pi%2F12-6pi%2F12=7pi%2F12-pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F12=7pi%2F12-6pi%2F12=7pi%2F12-pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I think we are expected to go to that table of trigonometric identities to find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29 ALIGN=MIDDLE  ALT=\"cos+%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Luckily, as everybody knows, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F2%29=0 ALIGN=MIDDLE  ALT=\"cos%28pi%2F2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F2%29=1 ALIGN=MIDDLE  ALT=\"sin%28pi%2F2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F12%29=cos%287pi%2F12-pi%2F2%29=cos%287pi%2F12%29cos%28pi%2F2%29%2Bsin%287pi%2F12%29sin%28pi%2F2%29+=+cos%287pi%2F12%29%2A0%2Bsin%287pi%2F12%29%2A1=sin%287pi%2F12%29=highlight%28%28sqrt%282%29%2Bsqrt%286%29%29%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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