document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=625300>Question 625300</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Tan 195 degrees  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #393450 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=393450\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that the instructions for the problem say to find the exact value. Otherwise we could just use our calculators and it wouldn't be much of a problem. (Please include the instructions so we don't have to guess.)<br><BR>\n" );
document.write( "Whenever the expression \"exact value\" is used in Trig. it is cods for \"use special angles\". 195 has a reference angle of 15 which is not a special angle. So 195 itself is not a special angle (which would have made this problem easy).<br><BR>\n" );
document.write( "Now we have to find a way to express 195 in terms of special angles. We have to ask ourselves:<ul><li>Is 195 the sum of two special angles?</li><li>Is 195 the difference of two special angles?</li><li>Is 195 the 1/2 of a special angle?</li><li>Is 195 the some other combination of special angles?</li></ul>If you're lucky, you will be able to answer \"yes\" to one of the first three questions. That is because we have formulas for sums, differences and halves of angles.<br><BR>\n" );
document.write( "With a little effort we should be able to answer \"yes\" to the first two equations:<BR>\n" );
document.write( "195 = 150 + 45<BR>\n" );
document.write( "and <BR>\n" );
document.write( "195 = 240 - 45<BR>\n" );
document.write( "So we will be able to get an answer in two ways:<BR>\n" );
document.write( "tan(195)<BR>\n" );
document.write( "tan(150+45)<BR>\n" );
document.write( "Using the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28A%2BB%29+=+%28tan%28A%29%2Btan%28B%29%29%2F%281-tab%28A%29%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A%2BB%29+=+%28tan%28A%29%2Btan%28B%29%29%2F%281-tab%28A%29%2Atan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tan%28150%29+%2B+tan%2845%29%29%2F%281-tan%28150%29%2Atan%2845%29%29 ALIGN=MIDDLE  ALT=\"%28tan%28150%29+%2B+tan%2845%29%29%2F%281-tan%28150%29%2Atan%2845%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We should be able to find these tan's easily since they are all special angles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-sqrt%283%29%2F3%29+%2B+%281%29%29%2F%281-%28-sqrt%283%29%2F3%29%2A%281%29%29 ALIGN=MIDDLE  ALT=\"%28%28-sqrt%283%29%2F3%29+%2B+%281%29%29%2F%281-%28-sqrt%283%29%2F3%29%2A%281%29%29\"   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=%28-sqrt%283%29%2F3+%2B+1%29%2F%281%2Bsqrt%283%29%2F3%29 ALIGN=MIDDLE  ALT=\"%28-sqrt%283%29%2F3+%2B+1%29%2F%281%2Bsqrt%283%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying the numerator and denominator by 3:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-sqrt%283%29+%2B+3%29%2F%283%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%28-sqrt%283%29+%2B+3%29%2F%283%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rationalizing the denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-sqrt%283%29+%2B+3%29%2F%283%2Bsqrt%283%29%29%29%2A%28%283-sqrt%283%29%29%2F%283-sqrt%283%29%29%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=%28%28-sqrt%283%29%29%2A3+%2B+%28-sqrt%283%29%29%2A%28-sqrt%283%29%29+%2B+3%2A3+%2B+3%2A%28-sqrt%283%29%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=%28-3sqrt%283%29+%2B+3+%2B+9+%2B+%28-3sqrt%283%29%29%29%2F%289-3%29 ALIGN=MIDDLE  ALT=\"%28-3sqrt%283%29+%2B+3+%2B+9+%2B+%28-3sqrt%283%29%29%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=%28-6sqrt%283%29+%2B+12%29%2F6 ALIGN=MIDDLE  ALT=\"%28-6sqrt%283%29+%2B+12%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=%286%28-sqrt%283%29+%2B+2%29%29%2F6 ALIGN=MIDDLE  ALT=\"%286%28-sqrt%283%29+%2B+2%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=%28cross%286%29%28-sqrt%283%29+%2B+2%29%29%2Fcross%286%29 ALIGN=MIDDLE  ALT=\"%28cross%286%29%28-sqrt%283%29+%2B+2%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=-sqrt%283%29+%2B+2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<br><BR>\n" );
document.write( "Another solution using 240 - 45:<BR>\n" );
document.write( "tan(195)<BR>\n" );
document.write( "tan(240-45)<BR>\n" );
document.write( "Using <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%2Btab%28A%29%2Atan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A-B%29+=+%28tan%28A%29-tan%28B%29%29%2F%281%2Btab%28A%29%2Atan%28B%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=%28tan%28240%29+-+tan%2845%29%29%2F%281%2Btan%28240%29%2Atan%2845%29%29 ALIGN=MIDDLE  ALT=\"%28tan%28240%29+-+tan%2845%29%29%2F%281%2Btan%28240%29%2Atan%2845%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=%28%28sqrt%283%29%29+-+%281%29%29%2F%281%2B%28sqrt%283%29%29%2A%281%29%29 ALIGN=MIDDLE  ALT=\"%28%28sqrt%283%29%29+-+%281%29%29%2F%281%2B%28sqrt%283%29%29%2A%281%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=%28sqrt%283%29+-+1%29%2F%281%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"%28sqrt%283%29+-+1%29%2F%281%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rationalizing the denoinator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28sqrt%283%29+-+1%29%2F%281%2Bsqrt%283%29%29%29%28%281-sqrt%283%29%29%2F%281-sqrt%283%29%29%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=%28sqrt%283%29%2A1+%2B+sqrt%283%29%2A%28-sqrt%283%29%29%2B%28-1%29%2A1%2B%28-1%29%28-sqrt%283%29%29%29%2F%28%281%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=%28sqrt%283%29+%2B+%28-3%29%2B%28-1%29%2Bsqrt%283%29%29%2F%281-3%29 ALIGN=MIDDLE  ALT=\"%28sqrt%283%29+%2B+%28-3%29%2B%28-1%29%2Bsqrt%283%29%29%2F%281-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29+%2B+%28-4%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"%282sqrt%283%29+%2B+%28-4%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2%29%28-sqrt%283%29+%2B+2%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"%28%28-2%29%28-sqrt%283%29+%2B+2%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%28%28-2%29%29%28-sqrt%283%29+%2B+2%29%29%2Fcross%28%28-2%29%29 ALIGN=MIDDLE  ALT=\"%28cross%28%28-2%29%29%28-sqrt%283%29+%2B+2%29%29%2Fcross%28%28-2%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=-sqrt%283%29+%2B+2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So either way we find that the exact value of tan(195) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29+%2B+2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );