document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074035>Question 1074035</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I find the exact value of the trigonometric expression given that cscU=-13/5 and cosV=3/4. How do I solve tan(V-U) using the given values?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #688858 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=688858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You do not seen to have quite enough information listed.<BR>\n" );
document.write( "With just the information you posted, the possibilities multiply, and you get 4 different answers.<BR>\n" );
document.write( "If you had some clue that led you to the signs (positive or negative) of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%29 ALIGN=MIDDLE  ALT=\"sin%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%29 ALIGN=MIDDLE  ALT=\"cos%28U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you could use the information given and trigonometric identities to find one answer.<BR>\n" );
document.write( "That one answer may start as a cumbersome expression requiring simplification,<BR>\n" );
document.write( "and ended as an ugly expression with square roots, but it would be the exact value.<BR>\n" );
document.write( "Without more information there are 2 possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%29 ALIGN=MIDDLE  ALT=\"sin%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and 2 possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%29 ALIGN=MIDDLE  ALT=\"cos%28U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , leading to 4 answers, and a lot more work.<BR>\n" );
document.write( "By definition, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28U%29=1%2Fsin%28U%29 ALIGN=MIDDLE  ALT=\"csc%28U%29=1%2Fsin%28U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28U%29=-13%2F5 ALIGN=MIDDLE  ALT=\"csc%28U%29=-13%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28U%29=-5%2F13 ALIGN=MIDDLE  ALT=\"sin%28U%29=-5%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be in quadrant 3, with a negative cosine,<BR>\n" );
document.write( "or in quadrant 4, with a positive cosine.<BR>\n" );
document.write( "Either way, we know that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > (as for any angle)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28U%29%2Bsin%5E2%28U%29=1 ALIGN=MIDDLE  ALT=\"cos%5E2%28U%29%2Bsin%5E2%28U%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28U%29=1-sin%5E2%28U%29=1-%28-5%2F13%29%5E2=1-25%2F169=%28169-25%29%2F169=144%2F169 ALIGN=MIDDLE  ALT=\"cos%5E2%28U%29=1-sin%5E2%28U%29=1-%28-5%2F13%29%5E2=1-25%2F169=%28169-25%29%2F169=144%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "meaning that either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%5B4%5D%29=sqrt%28144%2F169%29=12%2F13 ALIGN=MIDDLE  ALT=\"cos%28U%5B4%5D%29=sqrt%28144%2F169%29=12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%5B3%5D%29=-sqrt%28144%2F169%29=-12%2F13 ALIGN=MIDDLE  ALT=\"cos%28U%5B3%5D%29=-sqrt%28144%2F169%29=-12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Similarly, from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%29=3%2F4 ALIGN=MIDDLE  ALT=\"cos%28V%29=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28V%29=1-cos%5E2%28V%29=1-%283%2F4%29%5E2=1-9%2F16=7%2F16 ALIGN=MIDDLE  ALT=\"sin%5E2%28V%29=1-cos%5E2%28V%29=1-%283%2F4%29%5E2=1-9%2F16=7%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B1%5D%29=sqrt%287%2F16%29=sqrt%287%29%2F4 ALIGN=MIDDLE  ALT=\"sin%28V%5B1%5D%29=sqrt%287%2F16%29=sqrt%287%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5D ALIGN=MIDDLE  ALT=\"V%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > in quadrant 1,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B4%5D%29=-sqrt%287%2F16%29=-sqrt%287%29%2F4 ALIGN=MIDDLE  ALT=\"sin%28V%5B4%5D%29=-sqrt%287%2F16%29=-sqrt%287%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B4%5D ALIGN=MIDDLE  ALT=\"V%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > in quadrant 4.<BR>\n" );
document.write( " <BR>\n" );
document.write( "There are \"formulas\" called trigonometric identities<BR>\n" );
document.write( "that allow you to calculate the trigonometric functions of sums of angles, multiples of angles, and half of angles.<BR>\n" );
document.write( "You can look them up online;<BR>\n" );
document.write( "just search for \"trigonometric identities\" or \"trig identities.\"<BR>\n" );
document.write( "I see no point in memorizing most of those identities,<BR>\n" );
document.write( "unless your teacher/instructor insists that you must memorize them.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For this particular problem, there are different ways to go about it.<BR>\n" );
document.write( "1) You could use the trigonometric identities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29=sin%28A%29cos%28B%29-cos%28A%29sin%28B%29 ALIGN=MIDDLE  ALT=\"sin%28A-B%29=sin%28A%29cos%28B%29-cos%28A%29sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and/or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29 ALIGN=MIDDLE  ALT=\"cos%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V-U%29 ALIGN=MIDDLE  ALT=\"sin%28V-U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V-U%29 ALIGN=MIDDLE  ALT=\"cos%28V-U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then you could calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V-U%29=sin%28V-U%29%2Fcos%28V-U%29 ALIGN=MIDDLE  ALT=\"tan%28V-U%29=sin%28V-U%29%2Fcos%28V-U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "2) Alternatively, you could first calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%29=sin%28V%29%2Fcos%28V%29 ALIGN=MIDDLE  ALT=\"tan%28V%29=sin%28V%29%2Fcos%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%29=sin%28V%29%2Fcos%28V%29 ALIGN=MIDDLE  ALT=\"tan%28V%29=sin%28V%29%2Fcos%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and use them along with trig identity<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%29tan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A-B%29=%28tan%28A%29-tan%28B%29%29%2F%281%2Btan%28A%29tan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V-U%29 ALIGN=MIDDLE  ALT=\"tan%28V-U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Option 1) looks more complicated<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V-U%29=sin%28V%29cos%28U%29-cos%28V%29sin%28U%29=sin%28V%29cos%28U%29-%283%2F4%29%28-5%2F13%29=sin%28V%29cos%28U%29%2B15%2F52 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The four possibilities as to the signs (positive or negative) of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%29 ALIGN=MIDDLE  ALT=\"sin%28V%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%29 ALIGN=MIDDLE  ALT=\"cos%28U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "leads to either a positive or negative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%29cos%28U%29 ALIGN=MIDDLE  ALT=\"sin%28V%29cos%28U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B1%5D-U%5B4%5D%29=sin%28V%5B4%5D-U%5B3%5D%29=%28sqrt%287%29%2F4%29%2812%2F13%29%2B15%2F52=%2812sqrt%287%29%2B15%29%2F52 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%29cos%28U%29%3E0 ALIGN=MIDDLE  ALT=\"sin%28V%29cos%28U%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B1%5D-U%5B3%5D%29=sin%28V%5B4%5D-U%5B4%5D%29=-%28sqrt%287%29%2F4%29%2812%2F13%29%2B15%2F52=%2815-12sqrt%287%29%29%2F52 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > otherwise.<BR>\n" );
document.write( "For each of those sines, we could calculate two cosines as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28X%29=sqrt%281-sin%5E2%28X%29%29 ALIGN=MIDDLE  ALT=\"cos%28X%29=sqrt%281-sin%5E2%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28X%29=-sqrt%281-sin%5E2%28X%29%29 ALIGN=MIDDLE  ALT=\"cos%28X%29=-sqrt%281-sin%5E2%28X%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If that looks ugly, you could calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V-U%29 ALIGN=MIDDLE  ALT=\"cos%28V-U%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > directly from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A-B%29 ALIGN=MIDDLE  ALT=\"cos%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > trig identity.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V-U%29=cos%28V%29cos%28U%29%2Bsin%28V%29sin%28U%29=%283%2F4%29cos%28U%29%2Bsin%28V%29%28-5%2F13%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%5B4%5D-U%5B4%5D%29=%283%2F4%29%2812%2F13%29%2B%28sqrt%287%29%2F4%29%285%2F13%29=%2836%2B5sqrt%287%29%29%2F52 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%5B1%5D-U%5B4%5D%29=%2836-5sqrt%287%29%29%2F52 ALIGN=MIDDLE  ALT=\"cos%28V%5B1%5D-U%5B4%5D%29=%2836-5sqrt%287%29%29%2F52\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%5B1%5D-U%5B3%5D%29=-%2836%2B5sqrt%287%29%29%2F52=-%28cos%28V%5B4%5D-U%5B4%5D%29%29 ALIGN=MIDDLE  ALT=\"cos%28V%5B1%5D-U%5B3%5D%29=-%2836%2B5sqrt%287%29%29%2F52=-%28cos%28V%5B4%5D-U%5B4%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%5B4%5D-U%5B3%5D%29=-%2836-5sqrt%287%29%29%2F52=-%28cos%28V%5B1%5D-U%5B4%5D%29%29 ALIGN=MIDDLE  ALT=\"cos%28V%5B4%5D-U%5B3%5D%29=-%2836-5sqrt%287%29%29%2F52=-%28cos%28V%5B1%5D-U%5B4%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the four results are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D-U%5B4%5D%29=sin%28V%5B1%5D-U%5B4%5D%29%2Fcos%28V%5B1%5D-U%5B4%5D%29=%2812sqrt%287%29%2B15%29%2F%2836-5sqrt%287%29%29=%2812sqrt%287%29%2B15%29%2836%2B5sqrt%287%29%29%2F%28%2836-5sqrt%287%29%29%2836%2B5sqrt%287%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=tan%28V%5B1%5D-U%5B4%5D%29=%0D%0A%2836%2A12sqrt%287%29%2B12%2A5%2Asqrt%287%29%2Asqrt%287%29%2B15%2A36%2B15%2A5sqrt%287%29%29%2F%2836%5E2-%285sqrt%287%29%29%5E2%29=%0D%0A%28432sqrt%287%29%2B12%2A5%2A7%2B540%2B75sqrt%287%29%29%2F%281296-25%2A7%29=%0D%0A%28%28432%2B75%29sqrt%287%29%2B420%2B540%29%2F%281296-175%29=%0D%0Ahighlight%28%28507sqrt%287%29%2B960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D-U%5B3%5D%29=sin%28V%5B4%5D-U%5B3%5D%29%2Fcos%28V%5B4%5D-U%5B3%5D%29=sin%28V%5B1%5D-U%5B4%5D%29%2F%28-cos%28V%5B1%5D-U%5B4%5D%29%29=-tan%28V%5B1%5D-U%5B4%5D%29=highlight%28-%28507sqrt%287%29%2B960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D-U%5B4%5D%29=sin%28V%5B4%5D-U%5B4%5D%29%2Fcos%28V%5B4%5D-U%5B4%5D%29=%2815-12sqrt%287%29%29%2F%2836%2B5sqrt%287%29%29=%0D%0A%2815-12sqrt%287%29%29%2836-5sqrt%287%29%29%2F%28%2836%2B5sqrt%287%29%29%2836-5sqrt%287%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=tan%28V%5B4%5D-U%5B4%5D%29=%2815%2A36-15%2A5%2Asqrt%287%29-12%2A36%2Asqrt%287%29%2B12%2A5%2Asqrt%287%29%2Asqrt%287%29%29%2F1121=%0D%0A%28540-75sqrt%287%29-432sqrt%287%29%2B12%2A5%2A7%29%2F1121=%0D%0A%28540%2B420-%2875%2B432%29sqrt%287%29%29%2F1121=%0D%0Ahighlight%28%28960-507sqrt%287%29%29%2F1121=%28507sqrt%287%29-960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D-U%5B3%5D%29=sin%28V%5B1%5D-U%5B3%5D%29%2Fcos%28V%5B1%5D-U%5B3%5D%29=sin%28V%5B4%5D-U%5B4%5D%29%2F%28-cos%28V%5B4%5D-U%5B4%5D%29%29=-tan%28V%5B4%5D-U%5B4%5D%29=highlight%28%28507sqrt%287%29-960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Option 2):<BR>\n" );
document.write( "All the way at the top, we had<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%5B4%5D%29=12%2F13 ALIGN=MIDDLE  ALT=\"cos%28U%5B4%5D%29=12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28U%5B3%5D%29=-12%2F13 ALIGN=MIDDLE  ALT=\"cos%28U%5B3%5D%29=-12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" > paired with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28U%29=-5%2F13 ALIGN=MIDDLE  ALT=\"sin%28U%29=-5%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "for the two possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles.<BR>\n" );
document.write( "That makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28U%5B4%5D%29=-5%2F12 ALIGN=MIDDLE  ALT=\"tan%28U%5B4%5D%29=-5%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28U%5B3%5D%29=%28-5%29%2F%28-12%29=5%2F12 ALIGN=MIDDLE  ALT=\"tan%28U%5B3%5D%29=%28-5%29%2F%28-12%29=5%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "All the way at the top, we had<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B1%5D%29=sqrt%287%29%2F4 ALIGN=MIDDLE  ALT=\"sin%28V%5B1%5D%29=sqrt%287%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28V%5B4%5D%29=-sqrt%287%29%2F4 ALIGN=MIDDLE  ALT=\"sin%28V%5B4%5D%29=-sqrt%287%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > paired with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28V%29=3%2F4 ALIGN=MIDDLE  ALT=\"cos%28V%29=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the two possible <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles.<BR>\n" );
document.write( "That makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D%29=sqrt%287%29%2F3 ALIGN=MIDDLE  ALT=\"tan%28V%5B1%5D%29=sqrt%287%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D%29=-sqrt%287%29%2F3 ALIGN=MIDDLE  ALT=\"tan%28V%5B4%5D%29=-sqrt%287%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Plugging those values into the trig identity<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%29tan%28B%29%29 ALIGN=MIDDLE  ALT=\"tan%28A-B%29=%28tan%28A%29-tan%28B%29%29%2F%281%2Btan%28A%29tan%28B%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D-U%5B3%5D%29=%28tan%28V%5B1%5D%29-tan%28U%5B3%5D%29%29%2F%281%2Btan%28V%5B1%5D%29tan%28U%5B3%5D%29%29=%0D%0A%28sqrt%287%29%2F3-5%2F12%29%2F%281%2B%28sqrt%287%29%2F3%29%285%2F12%29%29=%28%2812sqrt%287%29-15%29%2F36%29%2F%281%2B5sqrt%287%29%2F36%29=%28%2812sqrt%287%29-15%29%2F36%29%2F%28%2836%2B5sqrt%287%29%29%2F36%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%28V%5B1%5D-U%5B3%5D%29=%2812sqrt%287%29-15%29%2F%2836%2B5sqrt%287%29%29=%0D%0A%2812sqrt%287%29-15%29%2836-5sqrt%287%29%29%2F%28%2836%2B5sqrt%287%29%29%2836-5sqrt%287%29%29%29=%0D%0A%2812%2A36%2Asqrt%287%29-12%2A5%2Asqrt%287%29%2Asqrt%287%29-36%2A15%2B15%2A5%2Asqrt%287%29%29%2F%2836%5E2-%285sqrt%287%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%28V%5B1%5D-U%5B3%5D%29=%28432sqrt%287%29-60%2A7-540%2B75sqrt%287%29%29%2F%281296-5%5E2%2A7%29=%0D%0A%28507sqrt%287%29-420-540%29%2F%281296-175%29=highlight%28%28507sqrt%287%29-960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D-U%5B4%5D%29=%28tan%28V%5B4%5D%29-tan%28U%5B4%5D%29%29%2F%281%2Btan%28V%5B4%5D%29tan%28U%5B4%5D%29%29%0D%0A=%28-sqrt%287%29%2F3-%28-5%2F12%29%29%2F%281%2B%28-sqrt%287%29%2F3%29%28-sqrt%287%29%2F3%29%29=%28sqrt%287%29%2F3-5%2F12%29%2F%281%2B%28sqrt%287%29%2F3%29%285%2F12%29%29=-tan%28V%5B1%5D-U%5B3%5D%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%28V%5B4%5D-U%5B4%5D%29=highlight%28-%28507sqrt%287%29-960%29%2F1121%29 ALIGN=MIDDLE  ALT=\"tan%28V%5B4%5D-U%5B4%5D%29=highlight%28-%28507sqrt%287%29-960%29%2F1121%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D-U%5B4%5D%29=%28tan%28V%5B1%5D%29-tan%28U%5B4%5D%29%29%2F%281%2Btan%28V%5B1%5D%29tan%28U%5B4%5D%29%29=%0D%0A%28sqrt%287%29%2F3-%28-5%2F12%29%29%2F%281%2B%28sqrt%287%29%2F3%29%28-5%2F12%29%29=%28sqrt%287%29%2F3%2B5%2F12%29%2F%281-%28sqrt%287%29%2F3%29%285%2F12%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=tan%28V%5B1%5D-U%5B4%5D%29=highlight%28%28507sqrt%287%29%2B960%29%2F1121%29 ALIGN=MIDDLE  ALT=\"tan%28V%5B1%5D-U%5B4%5D%29=highlight%28%28507sqrt%287%29%2B960%29%2F1121%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D%29 ALIGN=MIDDLE  ALT=\"tan%28V%5B4%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28U%5B3%5D%29 ALIGN=MIDDLE  ALT=\"tan%28U%5B3%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > both are the opposites<BR>\n" );
document.write( "of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B1%5D%29 ALIGN=MIDDLE  ALT=\"tan%28V%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28U%5B4%5D%29 ALIGN=MIDDLE  ALT=\"tan%28U%5B4%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > respectively,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28V%5B4%5D-U%5B3%5D%29=-tan%28V%5B1%5D-U%5B4%5D%29=highlight%28-%28507sqrt%287%29%2B960%29%2F1121%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );