document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=622504>Question 622504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me solve:\r<BR>\n" );
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document.write( "4tan^2u-1= tan^2u </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #391439 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=391439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28tan%28u%29%29%5E2-1=%28tan%28u%29%29%5E2 ALIGN=MIDDLE  ALT=\"4%28tan%28u%29%29%5E2-1=%28tan%28u%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let me change the name of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28u%29 ALIGN=MIDDLE  ALT=\"tan%28u%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a little while, to save ink and confusion.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28u%29=x%5E2 ALIGN=MIDDLE  ALT=\"tan%28u%29=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I can write the confusing equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-1=x%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2-1=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-1=0 ALIGN=MIDDLE  ALT=\"3x%5E2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2=1 ALIGN=MIDDLE  ALT=\"3x%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=1%2F3 ALIGN=MIDDLE  ALT=\"x%5E2=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Back into trigonometry,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tan%28u%29%29%5E2=1%2F3 ALIGN=MIDDLE  ALT=\"%28tan%28u%29%29%5E2=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You should know an angle that is a solution.<BR>\n" );
document.write( "(You should sine and cosine of 0, 30,45,60, and 90 degrees, or at least have a chart with their exact values handy).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%5Eo ALIGN=MIDDLE  ALT=\"30%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F6 ALIGN=MIDDLE  ALT=\"pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%5Eo%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"cos%2830%5Eo%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=tan%2830%5Eo%29=sin%2830%5Eo%29%2Fcos%2830%5Eo%29=1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%2830%5Eo%29=sin%2830%5Eo%29%2Fcos%2830%5Eo%29=1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(although teachers prefer the equivalent expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2830%5Eo%29=sqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"tan%2830%5Eo%29=sqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > because they do not like square roots in denominators).<BR>\n" );
document.write( "You also should know that the supplementary angle, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=150%5Eo ALIGN=MIDDLE  ALT=\"150%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F6 ALIGN=MIDDLE  ALT=\"5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the opposite tangent:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28150%5Eo%29=-1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%28150%5Eo%29=-1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So those two angles are solutions.<BR>\n" );
document.write( "There are many more solutions, because the tangent function has a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so to include all solutions we need to add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Api ALIGN=MIDDLE  ALT=\"n%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > defined as an integer to the solutions above.<BR>\n" );
document.write( "(I am not going to continue with the degrees, because I'm sure your teacher likes pies better than degrees).<BR>\n" );
document.write( "In radians, and putting all solutions in one formula, my solution set is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=n%2Api+%2B-+pi%2F6 ALIGN=MIDDLE  ALT=\"u=n%2Api+%2B-+pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all n integers </SPAN>\n" );
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