document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=634796>Question 634796</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I've been stuck on this one for a considerable amount of time.<BR>\n" );
document.write( "=> Show that the equation 4cos(2x)-3sin(x)csc(x)^3+6 = 0, can be expressed as <BR>\n" );
document.write( "8y^4-10y^2+3=0 if sin(x) = y<BR>\n" );
document.write( "Hence, solve the equation for x\r<BR>\n" );
document.write( "\n" );
document.write( "Thankyou for your help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #399927 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=399927\">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=4cos%282x%29-3sin%28x%29csc%5E3%28x%29%2B6+=+0 ALIGN=MIDDLE  ALT=\"4cos%282x%29-3sin%28x%29csc%5E3%28x%29%2B6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As the problem suggest, you want everything expressed in terms of sin(x). SO we will start by substituting for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29 ALIGN=MIDDLE  ALT=\"cos%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E3%28x%29 ALIGN=MIDDLE  ALT=\"csc%5E3%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The csc is easy. csc is the reciprocal of sin. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%5E3%28x%29+=+1%2Fsin%5E3%28x%29 ALIGN=MIDDLE  ALT=\"csc%5E3%28x%29+=+1%2Fsin%5E3%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We have three variations for cos(2x):<ul><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29+=+cos%5E2%28x%29-sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cos%282x%29+=+cos%5E2%28x%29-sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29+=+2cos%5E2%28x%29-1 ALIGN=MIDDLE  ALT=\"cos%282x%29+=+2cos%5E2%28x%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282x%29+=+1-2sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"cos%282x%29+=+1-2sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ul>Since we want everything in terms of sin(x), we will use the third variation. Substituting in we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281-2sin%5E2%28x%29%29-3sin%28x%29%281%2Fsin%5E3%28x%29%29%2B6+=+0 ALIGN=MIDDLE  ALT=\"4%281-2sin%5E2%28x%29%29-3sin%28x%29%281%2Fsin%5E3%28x%29%29%2B6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note the use of parentheses! These are important when making substitutions. Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-8sin%5E2%28x%29-%283%2Fsin%5E2%28x%29%29%2B6+=+0 ALIGN=MIDDLE  ALT=\"4-8sin%5E2%28x%29-%283%2Fsin%5E2%28x%29%29%2B6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8sin%5E2%28x%29-%283%2Fsin%5E2%28x%29%29%2B10+=+0 ALIGN=MIDDLE  ALT=\"-8sin%5E2%28x%29-%283%2Fsin%5E2%28x%29%29%2B10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now that everything is in terms of sin(x), the next task is to get rid of the fraction. Multiplying both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"sin%5E2%28x%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=-8sin%5E4%28x%29-3%2B10sin%5E2%28x%29+=+0 ALIGN=MIDDLE  ALT=\"-8sin%5E4%28x%29-3%2B10sin%5E2%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As you may see, we're getting close. We can get the signs right if we multiply both sides by -1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8sin%5E4%28x%29%2B3-10sin%5E2%28x%29+=+0 ALIGN=MIDDLE  ALT=\"8sin%5E4%28x%29%2B3-10sin%5E2%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And rearrange the terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8sin%5E4%28x%29-10sin%5E2%28x%29%2B3+=+0 ALIGN=MIDDLE  ALT=\"8sin%5E4%28x%29-10sin%5E2%28x%29%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can substitute \"y\" for \"sin(x)\":<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E4-10y%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"8y%5E4-10y%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The reason we made this substitution is that it is easier to see how to solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E4-10y%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"8y%5E4-10y%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "than it is to see how to solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8sin%5E4%28x%29-10sin%5E2%28x%29%2B3+=+0 ALIGN=MIDDLE  ALT=\"8sin%5E4%28x%29-10sin%5E2%28x%29%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Eventually you will see how to solve without making the substitution.) Note: An even better substitution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+sin%5E2%28x%29 ALIGN=MIDDLE  ALT=\"z+=+sin%5E2%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > This would make the substituted equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8z%5E2-10z%2B3=0 ALIGN=MIDDLE  ALT=\"8z%5E2-10z%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which I think you would agree is even easier to solve than:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E4-10y%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"8y%5E4-10y%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But we will go with the substitution your teacher/book suggested.<BR>\n" );
document.write( "<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E4-10y%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"8y%5E4-10y%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is an equation in what is called quadratic form. (A \"pure\" quadratic would have y-squared and y terms.) We can solve it with the same methods used to solve pure quadratics. One of these is factoring:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284y%5E2-3%29%282y%5E2-1%29+=+0 ALIGN=MIDDLE  ALT=\"%284y%5E2-3%29%282y%5E2-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If this factoring is not immediately obvious to you, then look at it for a while and let it sink in. Multiply it back out and see if you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E4-10y%5E2%2B3+=+0 ALIGN=MIDDLE  ALT=\"8y%5E4-10y%5E2%2B3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Look back at our alternate substitution, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8z%5E2-10z%2B3=0 ALIGN=MIDDLE  ALT=\"8z%5E2-10z%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and think about how that would factor. It might help you see how we got the factors above.<br><BR>\n" );
document.write( "Now we use the Zero Product Property:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2-3+=+0 ALIGN=MIDDLE  ALT=\"4y%5E2-3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2-1=0 ALIGN=MIDDLE  ALT=\"2y%5E2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving these...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2+=+3 ALIGN=MIDDLE  ALT=\"4y%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y%5E2=+1 ALIGN=MIDDLE  ALT=\"2y%5E2=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+3%2F4 ALIGN=MIDDLE  ALT=\"y%5E2+=+3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=+1%2F2 ALIGN=MIDDLE  ALT=\"y%5E2=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finding the square root of each side (and not forgetting about the negative square roots) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%2F4%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%2F4%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rationalizing the denominators...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29%2Fsqrt%284%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29%2Fsqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29%2Fsqrt%284%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29%2Fsqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28%281%2F2%29%282%2F2%29%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28%281%2F2%29%282%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%28%281%2F2%29%282%2F2%29%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%28%281%2F2%29%282%2F2%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=y+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%282%2F4%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%282%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%282%2F4%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%282%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%282%29%2Fsqrt%284%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%282%29%2Fsqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%282%29%2Fsqrt%284%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%282%29%2Fsqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We've solved for y. But we really want to solve for x. So it is time to substitute sin(x) back in for the y:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All of these are special angle sin's so every x is a special angle (so put your calculator away).<br><BR>\n" );
document.write( "We should recognize that the reference angle for both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And since we have both positive and negative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x will terminate in all four quadrants. So for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+-+pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+-+pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2B+pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+%2B+pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "with the middle two equations simplifying to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+2pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+4pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<br><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we should recognize that the reference angle will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x will again terminate in all four quadrants (since we have both positive and negative <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This gives us:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+-+pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+-+pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2B+pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi+%2B+pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "with the middle two equations simplifying to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+5pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<br><BR>\n" );
document.write( "The full general solution to your equation is all 8 of the equations above:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+2pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+4pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F3+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F3+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+3pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (2nd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+5pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (3rd quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-pi%2F4+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+-pi%2F4+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );