document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=261029>Question 261029</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to solve for x if 2sin^2x-3sinx=1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #192311 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=192311\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since the equation you posted has a terribly messy solution, I am going to assume that the equation is actually <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%5E2%28x%29-3sin%28x%29+=+-1 ALIGN=MIDDLE  ALT=\"2sin%5E2%28x%29-3sin%28x%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Would you know how to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%5E2+-+3q+=+-1 ALIGN=MIDDLE  ALT=\"2q%5E2+-+3q+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >? I ask because solving your equation involves<ul><li>Knowing how to solve my equation; and</li><li>Recognizing that your equation has the same structure as my equation.</li></ul><BR>\n" );
document.write( "First let's look at my equation. It is a quadratic equation. So we solve it by getting one side equal to zero:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%5E2+-+3q+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"2q%5E2+-+3q+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then factoring the left side (or using the Quadratic Formula). This factors fairly easily:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282q+-+1%29%28q+-+1%29+=+0 ALIGN=MIDDLE  ALT=\"%282q+-+1%29%28q+-+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By the Zero Product Property, this product is zero only if one of the factors is zero:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q+-+1+=+0 ALIGN=MIDDLE  ALT=\"2q+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+-+1+=+0 ALIGN=MIDDLE  ALT=\"q+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+1%2F2 ALIGN=MIDDLE  ALT=\"q+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+1 ALIGN=MIDDLE  ALT=\"q+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Next you need to see that your equation and mine have the same structure. They both say: \"2 times something squared minus 3 times that something equals -1\". If you still have trouble seeing this, then use a temporary variable:<BR>\n" );
document.write( "Let q = sin(x)<BR>\n" );
document.write( "Then substitute this variable into your equation for sin(x):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28q%29%5E2+-+3%2A%28q%29+=+-1 ALIGN=MIDDLE  ALT=\"2%28q%29%5E2+-+3%2A%28q%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is my equation! So you will end up with my solution:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+1%2F2 ALIGN=MIDDLE  ALT=\"q+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+1 ALIGN=MIDDLE  ALT=\"q+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Of course your are trying to solve for x, not the temporary variable q. So substitute sin(x) back in for q:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+1%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+1 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We still haven't solved for x but we're getting closer. Everything we have done so far is Algebra. At this point we finally need some Trig. We have to know when the sin function is 1/2 and when it is 1. If we know the special angles we know that for sin(x) = 1/2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F6+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F6+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5pi%2F6+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+5pi%2F6+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >}}}<BR>\n" );
document.write( "and for sin(x) = 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi%2F2+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"x+=+pi%2F2+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(Eventually you will learn how to solve problems like this without use of a temporary variable like q. But until then, feel free to use them knowing that at some point you will need to substitute back for it.) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );