document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1015621>Question 1015621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Solve for A[0<_A<_90°]<BR>\n" );
document.write( "TanA+Tan2A+Tan3A=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #631994 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=631994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> tan A + tan2A + tan3A = 0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tanA+%2B+tan2A%29+%2B+%28tanA+%2B+tan2A%29%2F%281-tanA%2Atan2A%29+=+0 ALIGN=MIDDLE  ALT=\"%28tanA+%2B+tan2A%29+%2B+%28tanA+%2B+tan2A%29%2F%281-tanA%2Atan2A%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=%28tanA+%2B+tan2A%29%2A%281+%2B+1%2F%281-tanA%2Atan2A%29%29+=+0 ALIGN=MIDDLE  ALT=\"%28tanA+%2B+tan2A%29%2A%281+%2B+1%2F%281-tanA%2Atan2A%29%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=%28tanA+%2B+tan2A%29%2A%28%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29%29+=+0 ALIGN=MIDDLE  ALT=\"%28tanA+%2B+tan2A%29%2A%28%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> tanA + tan2a = 0  OR <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29+=+0 ALIGN=MIDDLE  ALT=\"%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first case is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sinA%2Acos2A+%2B+sin2A%2AcosA%29%2F%28cosA%2Acos2A%29+=+0 ALIGN=MIDDLE  ALT=\"%28sinA%2Acos2A+%2B+sin2A%2AcosA%29%2F%28cosA%2Acos2A%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or equivalently, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin3A%29%2F%28cosA%2Acos2A%29+=+0 ALIGN=MIDDLE  ALT=\"%28sin3A%29%2F%28cosA%2Acos2A%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This happens only when 3A = 0, or A = 0 degree.  (Notice A = 0 does not make the denominator 0, so A = 0 is a valid solution.  It can be checked also by using the original equation that A = 0 is indeed a solution.)\r<BR>\n" );
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document.write( "The 2nd case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29+=+0 ALIGN=MIDDLE  ALT=\"%282+-+tanA%2Atan2A%29%2F%281-tanA%2Atan2A%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > implies that <BR>\n" );
document.write( "2 - tanA*tan2A = 0.  (Notice that this effectively means that the denominator 1 - tanA*tan2A is not equal to 0.)\r<BR>\n" );
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document.write( "==> tanA*tan2A = 2<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tanA%2A%28%282tanA%29%2F%281-%28tanA%29%5E2%29%29+=+2 ALIGN=MIDDLE  ALT=\"tanA%2A%28%282tanA%29%2F%281-%28tanA%29%5E2%29%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <==><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%28tanA%29%5E2%29%2F%281-%28tanA%29%5E2%29+=+2 ALIGN=MIDDLE  ALT=\"%282%28tanA%29%5E2%29%2F%281-%28tanA%29%5E2%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now the only value of A that will make the denominator of the last equation 0 is 45 degrees, in which tanA = 1.  Thus we exclude A = 45 degrees from the solution and <BR>\n" );
document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28tanA%29%5E2+=+2-2%28tanA%29%5E2 ALIGN=MIDDLE  ALT=\"2%28tanA%29%5E2+=+2-2%28tanA%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28tanA%29%5E2+=+2 ALIGN=MIDDLE  ALT=\"4%28tanA%29%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tanA%29%5E2+=+1%2F2 ALIGN=MIDDLE  ALT=\"%28tanA%29%5E2+=+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=tanA+=+1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"tanA+=+1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tanA+=+-1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"tanA+=+-1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The second instance is unacceptable, since it would situate A outside 0<_A<_90°.<BR>\n" );
document.write( "Thus we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+tan%5E-1%281%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"A+=+tan%5E-1%281%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Therefore the solution set for the original equation is {0, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%5E-1%281%2Fsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"tan%5E-1%281%2Fsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > } </SPAN>\n" );
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