document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=622815>Question 622815</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me find the exact values of:\r<BR>\n" );
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document.write( "tan2u\r<BR>\n" );
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document.write( "given sinu= -4/5 and pi< u < 3pi/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #391646 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=391646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is a formula that expresses tan(2u) in terms of tan(u):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282u%29+=+%282%2Atan%28u%29%29%2F%281-tan%5E2%28u%29%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%282%2Atan%28u%29%29%2F%281-tan%5E2%28u%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So if we can find tan(u) we can use this formula to find tan(2u).<br><BR>\n" );
document.write( "We are given that sin(u) = -4/5. We can use one of the Pythagorean identities, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28a%29+=+1-+sin%5E2%28a%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28a%29+=+1-+sin%5E2%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to use sin(u) to find cos(u). And once we have cos(u) we can use the fact that tan(u) = sin(u)/cos(u) to find tan(u).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28u%29+=+1+-+sin%5E2%28u%29 ALIGN=MIDDLE  ALT=\"cos%5E2%28u%29+=+1+-+sin%5E2%28u%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28u%29+=+1+-+%28-4%2F5%29%5E2 ALIGN=MIDDLE  ALT=\"cos%5E2%28u%29+=+1+-+%28-4%2F5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28u%29+=+1+-+16%2F25 ALIGN=MIDDLE  ALT=\"cos%5E2%28u%29+=+1+-+16%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28u%29+=+25%2F25+-+16%2F25 ALIGN=MIDDLE  ALT=\"cos%5E2%28u%29+=+25%2F25+-+16%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28u%29+=+9%2F25 ALIGN=MIDDLE  ALT=\"cos%5E2%28u%29+=+9%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we find the square root of each side. But which square root should we use? The positive one or the negative one? Well, since we are told that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%3C+u+%3C+3pi%2F2 ALIGN=MIDDLE  ALT=\"pi%3C+u+%3C+3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we know that u terminates in the 3rd quadrant. And in the 3rd quadrant cos is negative. So we should use the negative square root:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28u%29+=+-sqrt%289%2F25%29 ALIGN=MIDDLE  ALT=\"cos%28u%29+=+-sqrt%289%2F25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28u%29+=+-3%2F5 ALIGN=MIDDLE  ALT=\"cos%28u%29+=+-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can find tan(u):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28u%29+=+sin%28u%29%2Fcos%28u%29 ALIGN=MIDDLE  ALT=\"tan%28u%29+=+sin%28u%29%2Fcos%28u%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%28u%29+=+%28-4%2F5%29%2F%28-3%2F5%29 ALIGN=MIDDLE  ALT=\"tan%28u%29+=+%28-4%2F5%29%2F%28-3%2F5%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%28u%29+=+4%2F3 ALIGN=MIDDLE  ALT=\"tan%28u%29+=+4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can find tan(2u):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282u%29+=+%282%2Atan%28u%29%29%2F%281-tan%5E2%28u%29%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%282%2Atan%28u%29%29%2F%281-tan%5E2%28u%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%282u%29+=+%282%2A%284%2F3%29%29%2F%281-%284%2F3%29%5E2%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%282%2A%284%2F3%29%29%2F%281-%284%2F3%29%5E2%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%282u%29+=+%288%2F3%29%2F%281-%2816%2F9%29%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%288%2F3%29%2F%281-%2816%2F9%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%282u%29+=+%288%2F3%29%2F%289%2F9-%2816%2F9%29%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%288%2F3%29%2F%289%2F9-%2816%2F9%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%282u%29+=+%288%2F3%29%2F%28-7%2F9%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%288%2F3%29%2F%28-7%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When we divide fractions we change it to multiplying by the reciprocal:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282u%29+=+%288%2F3%29%2A%28-9%2F7%29 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+%288%2F3%29%2A%28-9%2F7%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%282u%29+=+-24%2F63 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+-24%2F63\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which reduces to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%282u%29+=+-8%2F21 ALIGN=MIDDLE  ALT=\"tan%282u%29+=+-8%2F21\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );