document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277603>Question 277603</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use an addition or subtraction formula to find the exact value of the expression. \r<BR>\n" );
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document.write( "Tan(-5pie/12)\r<BR>\n" );
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document.write( "The formula is tan(s-t)=tan s + tan t/ 1+tans tant<BR>\n" );
document.write( "or<BR>\n" );
document.write( "tan(s+t)= tan s-tan t/ 1+tans tant </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #202101 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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=202101\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Try:<BR>\n" );
document.write( "To make this more natural, remember that the primitive tangent function has period of pi. So we have Tan(-5pi/12) congruent to Tan(7pi/12) mod pi.\r<BR>\n" );
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document.write( "Now: What functions do we easily know the value to for trigonometrics? pi/3 is certainly one. So, pi/3=4pi/12---> We need to know 3pi/12=pi/4 ! This is also an easy find.\r<BR>\n" );
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document.write( "So, we take Tan(7pi/12)=Tan(3pi/12   +  4pi/12) Now we use your formulae:\r<BR>\n" );
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document.write( "Tan(3pi/12   +  4pi/12)=[Tan(pi/4)-Tan(pi/3)]/(1+Tan(pi/4)Tan(pi/3))\r<BR>\n" );
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document.write( "Now it remains to remember what Tan(pi/4) and Tan(pi/3) are. We use the definition of Tangent.<BR>\n" );
document.write( "Tan(x)=Sin(x)/Cos(x).<BR>\n" );
document.write( "So Tan(pi/4)=1.<BR>\n" );
document.write( "Tan(pi/3)=(sqrt(3)/2)/(1/2)=sqrt(3)\r<BR>\n" );
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document.write( "Finally, substitute into our expression:\r<BR>\n" );
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document.write( "(1-sqrt(3))/(1+sqrt(3))\r<BR>\n" );
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document.write( "Now, simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-sqrt%283%29%29%5E2%2F%281-3%29 ALIGN=MIDDLE  ALT=\"%281-sqrt%283%29%29%5E2%2F%281-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-2%2Asqrt%283%29%2B3%29%2A%281%2F%28-2%29%29 ALIGN=MIDDLE  ALT=\"%281-2%2Asqrt%283%29%2B3%29%2A%281%2F%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"-2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This could be done quicker, but I hope you can see the full details of what is going on through this procedure. </SPAN>\n" );
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