document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77922>Question 77922</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find all solutions for equation with value of x between 0 and 360 degrees:<BR>\n" );
document.write( "1.  sin(4x) + sin x = 0<BR>\n" );
document.write( "I have 2(2sin(x) cos (x))(cox(2x)) + sin x = 0<BR>\n" );
document.write( "so far but don't know which identity to use for cos(2x) that will help solve.\r<BR>\n" );
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document.write( "This is from my daughter's textbook but I don't have the book with me now. She's taking pre-calc in high school. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55857 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=55857\">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=sin%284x%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"sin%284x%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282%2A%282x%29%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"sin%282%2A%282x%29%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%284x%29 ALIGN=MIDDLE  ALT=\"sin%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282%2A%282x%29%29 ALIGN=MIDDLE  ALT=\"sin%282%2A%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%282x%29cos%282x%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"2sin%282x%29cos%282x%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2x=2sinxcosx ALIGN=MIDDLE  ALT=\"sin2x=2sinxcosx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282sin%28x%29cos%28x%29%29cos%282x%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"2%282sin%28x%29cos%28x%29%29cos%282x%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2x=2sinxcosx ALIGN=MIDDLE  ALT=\"sin2x=2sinxcosx\"   DISABLED_style= \"max-width:90%; height:auto;\" > again\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284sin%28x%29cos%28x%29%29%282%28cos%28x%29%29%5E2-1%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"%284sin%28x%29cos%28x%29%29%282%28cos%28x%29%29%5E2-1%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the identity: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos2x=2%28cos%28x%29%29%5E2-1 ALIGN=MIDDLE  ALT=\"cos2x=2%28cos%28x%29%29%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8sin%28x%29%28cos%28x%29%29%5E3-4sin%28x%29cos%28x%29+%2B+sin+x+=+0 ALIGN=MIDDLE  ALT=\"8sin%28x%29%28cos%28x%29%29%5E3-4sin%28x%29cos%28x%29+%2B+sin+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sin%28x%29cos%28x%29 ALIGN=MIDDLE  ALT=\"4sin%28x%29cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%288%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1%29+=+0 ALIGN=MIDDLE  ALT=\"sin%28x%29%288%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out a sin(x)\r<BR>\n" );
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document.write( "Since we know the value of x for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the solution is x=0) we can ignore the sin(x) and try to solve the expression in the parenthesis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"8%28cos%28x%29%29%5E3-4cos%28x%29+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > So lets focus on the terms in the parenthesis\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=cos%28x%29 ALIGN=MIDDLE  ALT=\"y=cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3-4y%2B1=0 ALIGN=MIDDLE  ALT=\"8y%5E3-4y%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3-2y-2y%2B1=0 ALIGN=MIDDLE  ALT=\"8y%5E3-2y-2y%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite -4y into -2y-2y. This will allow us to factor\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%284y%5E2-1%29-%282y-1%29=0 ALIGN=MIDDLE  ALT=\"2y%284y%5E2-1%29-%282y-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms and factor out the GCF\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%282y%2B1%29%282y-1%29-%282y-1%29=0 ALIGN=MIDDLE  ALT=\"2y%282y%2B1%29%282y-1%29-%282y-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2-1 ALIGN=MIDDLE  ALT=\"4y%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282y%2B1%29%282y-1%29 ALIGN=MIDDLE  ALT=\"%282y%2B1%29%282y-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using difference of squares\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282y%282y%2B1%29-1%29%282y-1%29=0 ALIGN=MIDDLE  ALT=\"%282y%282y%2B1%29-1%29%282y-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms (note: the common term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-1 ALIGN=MIDDLE  ALT=\"2y-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Now set each factor equal to zero. Lets start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-1 ALIGN=MIDDLE  ALT=\"2y-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-1=0 ALIGN=MIDDLE  ALT=\"2y-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y=1 ALIGN=MIDDLE  ALT=\"2y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F2 ALIGN=MIDDLE  ALT=\"y=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28x%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for x  note: I'm using radians\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.04719%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.04719%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.04719%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=-1.04719%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.04719%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.04719%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only solution.\r<BR>\n" );
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document.write( "Now let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%282y%2B1%29-1=0 ALIGN=MIDDLE  ALT=\"2y%282y%2B1%29-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B2y-1=0 ALIGN=MIDDLE  ALT=\"4y%5E2%2B2y-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the 2y\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for y\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ay%5E2%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B2y%2B-1+=+0 ALIGN=MIDDLE  ALT=\"4y%5E2%2B2y%2B-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%282%29%5E2-4%2A4%2A-1=20 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%282%29%5E2-4%2A4%2A-1=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=20 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-2%2B-sqrt%28+20+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-2%2B-sqrt%28+20+%29%29%2F2%5Ca\"   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=y%5B1%5D+=+%28-%282%29%2Bsqrt%28+20+%29%29%2F2%5C4+=+0.309016994374947 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+%28-%282%29%2Bsqrt%28+20+%29%29%2F2%5C4+=+0.309016994374947\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-%282%29-sqrt%28+20+%29%29%2F2%5C4+=+-0.809016994374947 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-%282%29-sqrt%28+20+%29%29%2F2%5C4+=+-0.809016994374947\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B2y%2B-1 ALIGN=MIDDLE  ALT=\"4y%5E2%2B2y%2B-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B2y%2B-1+=+4%28y-0.309016994374947%29%2A%28y--0.809016994374947%29 ALIGN=MIDDLE  ALT=\"4y%5E2%2B2y%2B-1+=+4%28y-0.309016994374947%29%2A%28y--0.809016994374947%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.309016994374947, -0.809016994374947.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B2%2Ax%2B-1+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B2%2Ax%2B-1+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "So if we replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get these solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=0.309017 ALIGN=MIDDLE  ALT=\"cos%28x%29=0.309017\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=-0.809017 ALIGN=MIDDLE  ALT=\"cos%28x%29=-0.809017\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take the arccosine of both sides (for the solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=0.309017 ALIGN=MIDDLE  ALT=\"cos%28x%29=0.309017\"   DISABLED_style= \"max-width:90%; height:auto;\" >) to solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.25664%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.25664%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.25664%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=-1.25664%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > Here are 2 more possible solutions. Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.25664%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.25664%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only solution.\r<BR>\n" );
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document.write( "Now lets use the other answer of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=-0.809017 ALIGN=MIDDLE  ALT=\"cos%28x%29=-0.809017\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=-0.809017 ALIGN=MIDDLE  ALT=\"cos%28x%29=-0.809017\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take the arccosine of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.51327%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=2.51327%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2.51327%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=-2.51327%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since our interval is [0,2pi] we must ignore the negative answer. So of these two answers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.51327%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=2.51327%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only solution.\r<BR>\n" );
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document.write( "So after all of that, we find that our solutions are\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (from sin(x)=0) or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.04719%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.04719%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.25664%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=1.25664%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.51327%2Bpi%2An ALIGN=MIDDLE  ALT=\"x=2.51327%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "As always, we can check our work by using a calculator.\r<BR>\n" );
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document.write( "This is a lot to take in, so feel free to ask me further about any of this. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );