document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=815617>Question 815617</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve 3sin^2(w)-17sin(w)+10=0 for all solutions where w is in radians between 0 and 2pi </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #491179 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=491179\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: When I first posted a solution, I had neglected to set my calculator to radian mode. So there were errors in the solution. This posting has these errors corrected.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sin%5E2%28w%29-17sin%28w%29%2B10=0 ALIGN=MIDDLE  ALT=\"3sin%5E2%28w%29-17sin%28w%29%2B10=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First we factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283sin%28w%29-2%29%28sin%28w%29-5%29=0 ALIGN=MIDDLE  ALT=\"%283sin%28w%29-2%29%28sin%28w%29-5%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If you cannot see how the equation factored this way, then try using a temporary variable. Let q = sin(w). Then the equation becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3q%5E2-17q%2B10=0 ALIGN=MIDDLE  ALT=\"3q%5E2-17q%2B10=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. It should not be too difficult to see that it factors into (3q-2)(q-5). Then just replace the q's.<br><BR>\n" );
document.write( "Next we use the Zero Product Property:<BR>\n" );
document.write( "3sin(w)-2 = 0 or sin(w)-5=0<BR>\n" );
document.write( "Solving 3sin(w)-2 = 0 ...<BR>\n" );
document.write( "3sin(w) = 2<BR>\n" );
document.write( "sin(w) = 2/3<BR>\n" );
document.write( "Using inverse sin we get a reference angle of (approximately) 0.73. Since the 2/3 is positive and since sin is positive in the 1st and 2nd quadrants we get a general solution of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+0.73+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"w+=+0.73+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for 1st quadrant angles)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+pi-0.73+%2B+2pi%2An+=+2.41+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"w+=+pi-0.73+%2B+2pi%2An+=+2.41+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for 2nd quadrant angles)<br><BR>\n" );
document.write( "Solving sin(w) - 5 - 0 ...<BR>\n" );
document.write( "sin(w) = 5<BR>\n" );
document.write( "But sin is never larger than 1 (or less than -1). So there is no solution for this.<br><BR>\n" );
document.write( "Last we use the general solution equations to find the specific solutions in the specified interval. <BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+0.73+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"w+=+0.73+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if n = 0 then w = 0.73<BR>\n" );
document.write( "if n = 1 (or larger) then w is too large for the interval<BR>\n" );
document.write( "if n = -1 (or smaller) then w is too small for the interval<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+2.41+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"w+=+2.41+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for 2nd quadrant angles)<br><BR>\n" );
document.write( "if n = 0 then w = 2.41<BR>\n" );
document.write( "if n = 1 (or larger) then w is too large for the interval<BR>\n" );
document.write( "if n = -1 (or smaller) then w is too small for the interval<br><BR>\n" );
document.write( "So the only solutions in the given interval are 0.73 and 2.41. </SPAN>\n" );
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