document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=421010>Question 421010</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the sum or difference to find the exact value of tan105 degrees.\r<BR>\n" );
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document.write( "please help? Thank u. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #294039 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</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=294039\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Whenever the phrase \"exact value\" is used in referring to the value of a trig function, it means that you must use special angles because exact values are only available for the special angles.<br><BR>\n" );
document.write( "The special angles are all the integer multiples of 30 and 45 degrees. The exact value for any of these angles can be found.<br><BR>\n" );
document.write( "In your problem, 105 degrees is not a multiple of 30 or 45 so it is not one of the special angles. But, as the problem mentions, it is possible to express 105 as the sum or difference of special angles. There are many possible sums or differences of special angles that result in 105. One of them is:<BR>\n" );
document.write( "135 - 30 = 105<BR>\n" );
document.write( "135 is special because it is 3*45 and 30 special and this difference of special angles results in 105. So we will use this difference and the formula for the tan of a difference:<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%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A-B%29+=+%28tan%28A%29+-+tan%28B%29%29%2F%281%2Btan%28A%29%2Atan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using 135 for A and 30 for B in this formula we getL<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28135-30%29+=+%28tan%28135%29+-+tan%2830%29%29%2F%281%2Btan%28135%29%2Atan%2830%29%29 ALIGN=MIDDLE  ALT=\"tan%28135-30%29+=+%28tan%28135%29+-+tan%2830%29%29%2F%281%2Btan%28135%29%2Atan%2830%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The left side simplifies to tan(105). On the right side we find the exact values for the tan of the special angles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28%28-1%29+-+%28sqrt%283%29%2F3%29%29%2F%281%2B%28-1%29%2A%28sqrt%283%29%2F3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To eliminate the fractions within the fraction on the right side, I am going to multiply the numerator and denominator by 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28%28%28-1%29+-+%28sqrt%283%29%2F3%29%29%2F%281%2B%28-1%29%2A%28sqrt%283%29%2F3%29%29%29%283%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28-3+-+sqrt%283%29%29%2F%283%2B%28-sqrt%283%29%29%29 ALIGN=MIDDLE  ALT=\"tan%28105%29+=+%28-3+-+sqrt%283%29%29%2F%283%2B%28-sqrt%283%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And if we want a rational denominator we need to multiply the numerator and denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"3%2Bsqrt%283%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%28105%29+=+%28%28-3+-+sqrt%283%29%29%2F%283%2B%28-sqrt%283%29%29%29%29%28%283%2Bsqrt%283%29%29%2F%283%2Bsqrt%283%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28-9+%2B+%28-3sqrt%283%29%29+%2B+%28-3sqrt%283%29%29+%2B+%28-3%29%29%2F%28%283%29%5E2+-+%28sqrt%283%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28-12+%2B+%28-6sqrt%283%29%29%29%2F%289+-+3%29%29%29%29%0D%0A%7B%7B%7Btan%28105%29+=+%28-12+%2B+%28-6sqrt%283%29%29%29%2F6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%286%28-2+%2B+%28-sqrt%283%29%29%29%29%2F6 ALIGN=MIDDLE  ALT=\"tan%28105%29+=+%286%28-2+%2B+%28-sqrt%283%29%29%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28105%29+=+%28cross%286%29%28-2+-+sqrt%283%29%29%29%2Fcross%286%29 ALIGN=MIDDLE  ALT=\"tan%28105%29+=+%28cross%286%29%28-2+-+sqrt%283%29%29%29%2Fcross%286%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%28105%29+=+-2+-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%28105%29+=+-2+-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the simplified, exact value for tan(105). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );