document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207450>Question 207450</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the exact value of tan 15degrees without using tables or a calculator </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156871 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=156871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The only \"exact values\" for Trigonometric functions which can be found are for 0, 30, 45, 60, 90 degress (and for angles whose reference angles are these). So the only way to find the excat value of tan(15) is to express the tan(15) in terms of these other angles.\r<BR>\n" );
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document.write( "Since 15 = 1/2 of 30 we could use the 1/2 angle formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%281%2F2%29x+=+%281+-+cos%28x%29%29%2F%28sin%28x%29%29 ALIGN=MIDDLE  ALT=\"tan%281%2F2%29x+=+%281+-+cos%28x%29%29%2F%28sin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where x = 30:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%281%2F2%2930+=+%281+-+cos%2830%29%29%2F%28sin%2830%29%29 ALIGN=MIDDLE  ALT=\"tan%281%2F2%2930+=+%281+-+cos%2830%29%29%2F%28sin%2830%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%29+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%2830%29+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%29+=+1%2F2 ALIGN=MIDDLE  ALT=\"sin%2830%29+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%28%281+-+sqrt%283%29%2F2%29%29%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%28%281+-+sqrt%283%29%2F2%29%29%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Change the division by 1/2 to mutiplying by 2/1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%281+-+sqrt%283%29%29+%2A+%282%2F1%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%281+-+sqrt%283%29%29+%2A+%282%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+2+-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+2+-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Another way would be to use the fact that 15 = 45 - 30 and use the difference formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28a-b%29+=+%28tan%28a%29+-+tan%28b%29%29%2F%281%2Btan%28a%29+tan%28b%29%29 ALIGN=MIDDLE  ALT=\"tan%28a-b%29+=+%28tan%28a%29+-+tan%28b%29%29%2F%281%2Btan%28a%29+tan%28b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with a = 45 and b = 30:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2845-30%29+=+%28tan%2845%29+-+tan%2830%29%29%2F%281+%2B+tan%2845%29+tan%2830%29%29 ALIGN=MIDDLE  ALT=\"tan%2845-30%29+=+%28tan%2845%29+-+tan%2830%29%29%2F%281+%2B+tan%2845%29+tan%2830%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2845%29+=+1 ALIGN=MIDDLE  ALT=\"tan%2845%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2830%29+=+sqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"tan%2830%29+=+sqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%281+-+sqrt%283%29%2F3%29%2F%281+%2B+1+%2A+%28sqrt%283%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%281+-+sqrt%283%29%2F3%29%2F%281+%2B+1+%2A+%28sqrt%283%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying the top and bottom by 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%283+-+sqrt%283%29%29%2F%283+%2B+sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%283+-+sqrt%283%29%29%2F%283+%2B+sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we rationalize the denominator by mulitplying by its conjugate: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283+-+sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%283+-+sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%289+-+6%2Asqrt%283%29+%2B+3%29%2F%289+-+3%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%289+-+6%2Asqrt%283%29+%2B+3%29%2F%289+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%2812+-+6%2Asqrt%283%29%29%2F6 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%2812+-+6%2Asqrt%283%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring out 6 on top:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+%286%282+-+sqrt%283%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+%286%282+-+sqrt%283%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Canceling the common factor of 6:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%29+=+2+-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%2815%29+=+2+-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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