document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114238>Question 114238</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) use Euler's formula to determine the standard trigonometric identity for cos(alpha+beta). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83140 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=83140\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Euler's Formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28ix%29=cos%28x%29%2Bi%2Asin%28x%29 ALIGN=MIDDLE  ALT=\"e%5E%28ix%29=cos%28x%29%2Bi%2Asin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Apply this formula to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=alpha%2Bbeta ALIGN=MIDDLE  ALT=\"x=alpha%2Bbeta\"   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=e%5E%28i%28alpha%2Bbeta%29%29=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29 ALIGN=MIDDLE  ALT=\"e%5E%28i%28alpha%2Bbeta%29%29=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28i%28alpha%2Bbeta%29%29=e%5E%28i%2Aalpha%29e%5E%28i%2Abeta%29 ALIGN=MIDDLE  ALT=\"e%5E%28i%28alpha%2Bbeta%29%29=e%5E%28i%2Aalpha%29e%5E%28i%2Abeta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28i%28alpha%2Bbeta%29%29=%28cos%28alpha%29%2Bi%2Asin%28alpha%29%29%28cos%28beta%29%2Bi%2Asin%28beta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which is to say, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29=%28cos%28alpha%29%2Bi%2Asin%28alpha%29%29%28cos%28beta%29%2Bi%2Asin%28beta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Expanding the right side using FOIL yields:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29%2Bcos%28alpha%29%2Ai%2Asin%28beta%29%2Bi%2Asin%28alpha%29cos%28beta%29%2B%28i%5E2%29sin%28alpha%29sin%28beta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And gathering like terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29%2Bi%28cos%28alpha%29sin%28beta%29%2Bsin%28alpha%29cos%28beta%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi=c%2Bdi ALIGN=MIDDLE  ALT=\"a%2Bbi=c%2Bdi\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=c ALIGN=MIDDLE  ALT=\"a=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=d ALIGN=MIDDLE  ALT=\"b=d\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "we can see that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29 ALIGN=MIDDLE  ALT=\"cos%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the desired identity.\r<BR>\n" );
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document.write( "By the way, we have also proven the companion identity:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%2Bbeta%29=cos%28alpha%29sin%28beta%29%2Bsin%28alpha%29cos%28beta%29 ALIGN=MIDDLE  ALT=\"sin%28alpha%2Bbeta%29=cos%28alpha%29sin%28beta%29%2Bsin%28alpha%29cos%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hope this helps,<BR>\n" );
document.write( "John<BR>\n" );
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