document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515214>Question 515214</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use a half-angle formula to find the exact value of tan 3pi/8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #343821 by </B><A HREF=/tutors/aboutme.mpl?userid=drcole><B>drcole(72)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drcole><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=343821\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The half angle formula for tangent can be written in two different but equally simple ways:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan+%28theta%2F2%29+=+%281+-+cos+%28theta%29%29%2F%28sin+%28theta%29%29+=+%28sin+%28theta%29%29%2F%281+%2B+cos+%28theta%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We'll use the first formula, but you could just as easily use the second and get the same answer.  We want to find the tangent of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29+pi ALIGN=MIDDLE  ALT=\"%283%2F8%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29+pi ALIGN=MIDDLE  ALT=\"%283%2F8%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one half of the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F4%29+pi ALIGN=MIDDLE  ALT=\"%283%2F4%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, since:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28%283%2F4%29+pi%29%2F+2+=+%281%2F2%29+%2A+%283%2F4%29+pi+=+%283%2F8%29+pi ALIGN=MIDDLE  ALT=\"+%28%283%2F4%29+pi%29%2F+2+=+%281%2F2%29+%2A+%283%2F4%29+pi+=+%283%2F8%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we'll try applying the half angle formula for tangent using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+%283%2F4%29+pi ALIGN=MIDDLE  ALT=\"theta+=+%283%2F4%29+pi\"   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=+tan+%28%283%2F8%29+pi%29+=+tan+%28%28%283%2F4%29+pi%29%2F+2%29+=+%281+-+cos+%28%283%2F4%29+pi%29%29%2F+%28sin+%28%283%2F4%29+pi%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Using the unit circle, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+%28%283%2F4%29+pi%29+=+-+%28sqrt%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"cos+%28%283%2F4%29+pi%29+=+-+%28sqrt%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%28%283%2F4%29+pi%29+=+%28sqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"sin+%28%283%2F4%29+pi%29+=+%28sqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this makes sense since the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F4%29+pi ALIGN=MIDDLE  ALT=\"%283%2F4%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the second quadrant and makes a 45 degree angle with the negative x-axis).  So we substitute in:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+-+cos+%28%283%2F4%29+pi%29%29%2F+%28sin+%28%283%2F4%29+pi%29%29+=+%281+-+%28-%28sqrt%282%29%29%2F2%29%29%2F%28sqrt%282%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplifying, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+%2B+%28sqrt%282%29%29%2F2%29%2F%28%28sqrt%282%29%29%2F2%29+ ALIGN=MIDDLE  ALT=\"%281+%2B+%28sqrt%282%29%29%2F2%29%2F%28%28sqrt%282%29%29%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (canceling the negatives in the numerator)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F%28sqrt%282%29%29+%2A+%281+%2B+%28sqrt%282%29%29%2F2%29%29+ ALIGN=MIDDLE  ALT=\"+%282%2F%28sqrt%282%29%29+%2A+%281+%2B+%28sqrt%282%29%29%2F2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (getting rid of the complex fraction by multiplying by the reciprocal of the denominator instead)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2F%28sqrt%282%29%29+%2B+1+ ALIGN=MIDDLE  ALT=\"+2%2F%28sqrt%282%29%29+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (distributing the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"2%2F%28sqrt%282%29%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=+sqrt%282%29+%2B+1+ ALIGN=MIDDLE  ALT=\"+sqrt%282%29+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (simplifying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"2%2F%28sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So the tangent of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29+pi ALIGN=MIDDLE  ALT=\"%283%2F8%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+%2B+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > exactly.  This should match what you get on your calculator, and it also makes sense: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29+pi ALIGN=MIDDLE  ALT=\"%283%2F8%29+pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the first quadrant, so its tangent should be positive, and it is larger angle than 45 degrees, so its tangent should be larger than 1. </SPAN>\n" );
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