document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823002>Question 823002</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve for theta where 0<theta<2pi<BR>\n" );
document.write( "I) 2sin2theta= -1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495199 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=495199\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: After seeing your thank you note I came back to my solution and checked. I'm sorry to say that I was wrong about the reference angle. A reference angle of <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;\" > not <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;\" > has a sin of 1/2. Everything else was correct. The problem below has been redone using the proper reference angle.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%282theta%29+=+-1 ALIGN=MIDDLE  ALT=\"2sin%282theta%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First isolate the function and its argument by dividing both sides by 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282theta%29+=+-1%2F2 ALIGN=MIDDLE  ALT=\"sin%282theta%29+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We should recognize that 1/2 (positive or negative) is a special angle value for sin. Without our calculator we should know that a 1/2 for sin indicates a reference angle of <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;\" >. Since the 1/2 is negative and since sin is negative in the 3rd and 4th quadrants we should get the following general solution equations (for 2 theta):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta+=+pi%2Bpi%2F6%2B2pi%2An ALIGN=MIDDLE  ALT=\"2theta+=+pi%2Bpi%2F6%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 3rd quadrant)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta+=+-pi%2F6%2B2pi%2An ALIGN=MIDDLE  ALT=\"2theta+=+-pi%2F6%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the 4th quadrant)<BR>\n" );
document.write( "The first equation will simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta+=+7pi%2F6%2B2pi%2An ALIGN=MIDDLE  ALT=\"2theta+=+7pi%2F6%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2theta+=+-pi%2F6%2B2pi%2An ALIGN=MIDDLE  ALT=\"2theta+=+-pi%2F6%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides of both equations by 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+7pi%2F12%2Bpi%2An ALIGN=MIDDLE  ALT=\"theta+=+7pi%2F12%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+-pi%2F12%2Bpi%2An ALIGN=MIDDLE  ALT=\"theta+=+-pi%2F12%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we try various integer values for n as we search for specific solutions which are in the specified interval.<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+7pi%2F12%2Bpi%2An ALIGN=MIDDLE  ALT=\"theta+=+7pi%2F12%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "If n = 0 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+7pi%2F12 ALIGN=MIDDLE  ALT=\"theta+=+7pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 1 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+19pi%2F12 ALIGN=MIDDLE  ALT=\"theta+=+19pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 2 (or larger) then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is too large for the interval<BR>\n" );
document.write( "If n = -1 (or smaller) then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is too small for the interval<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+-pi%2F12%2Bpi%2An ALIGN=MIDDLE  ALT=\"theta+=+-pi%2F12%2Bpi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "If n = 0 (or smaller) then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is too small for the interval<BR>\n" );
document.write( "If n = 1 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+11pi%2F12 ALIGN=MIDDLE  ALT=\"theta+=+11pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 2 then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+23pi%2F12 ALIGN=MIDDLE  ALT=\"theta+=+23pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If n = 3 (or larger) then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is too large for the interval<br><BR>\n" );
document.write( "So the only solutions in the specified interval are: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7pi%2F12 ALIGN=MIDDLE  ALT=\"7pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19pi%2F12 ALIGN=MIDDLE  ALT=\"19pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11pi%2F12 ALIGN=MIDDLE  ALT=\"11pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23pi%2F12 ALIGN=MIDDLE  ALT=\"23pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );