document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155366>Question 1155366</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me to answer this:\r<BR>\n" );
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document.write( "For 0° ≤  x  ≤  360°, find the number of roots the equation 2sinxtanx = -tanx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777946 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=777946\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe this will have 5 roots between 0 and 360 degrees.<BR>\n" );
document.write( "the roots will be at 0, 180, 360 degrees for tan(x) = 0, and 210, 330 degrees for sin(x) = -1/2.<BR>\n" );
document.write( "to confirm, graph the equation as shown below:\r<BR>\n" );
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document.write( "to solve algebraically, do the following.\r<BR>\n" );
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document.write( "start with 2 * sin(x) * tan(x) = -tan(x)<BR>\n" );
document.write( "add tan(x) to both sides of the equation to get:<BR>\n" );
document.write( "2 * sin(x) * tan(x) + tan(x) = 0<BR>\n" );
document.write( "factor out tan(x) to get:<BR>\n" );
document.write( "tan(x) * (2 * sin(x) + 1) = 0<BR>\n" );
document.write( "this is true when tan(x) = 0 or sin(x) + 1 = 0 <BR>\n" );
document.write( "when tan(x) = 0, x = 0 or 180 or 360 degrees.<BR>\n" );
document.write( "when 2 * sin(x) + 1 = 0, solve for sin(x) to get:<BR>\n" );
document.write( "sin(x) = -1/2<BR>\n" );
document.write( "solve for sin(x) = plus 1/2 to get x = 30 degrees.<BR>\n" );
document.write( "that's in the first quadrant where all trig functions are positive and where all reference angles reside.<BR>\n" );
document.write( "sine is negative in the third and fourth quadrants.<BR>\n" );
document.write( "in the third quadrant 180 + 30 = 210 degrees.<BR>\n" );
document.write( "in the fourth quadrant 360 - 30 = 330 degrees.<BR>\n" );
document.write( "confirm by solving for sin in all 4 quadrants to see that the sine is only negative in the third and fourth quadrant.<BR>\n" );
document.write( "30 degrees in the second quadrant = 180 - 30 = 150.<BR>\n" );
document.write( "30 degrees in the third quadrant = 180 + 30 = 210.<BR>\n" );
document.write( "30 degrees in the fourth quadrant = 360 - 30 = 330 degrees.<BR>\n" );
document.write( "sin(30) = 1/2<BR>\n" );
document.write( "sin(150) = 1/2<BR>\n" );
document.write( "sin(210) = -1/2<BR>\n" );
document.write( "sin(330) = -1/2<BR>\n" );
document.write( "the sine is -1/2 at 210 and 330 degrees only in the interval between 0 and 360 degrees.\r<BR>\n" );
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document.write( "bottom line:<BR>\n" );
document.write( "5 roots.<BR>\n" );
document.write( "0, 180, 360 for tan(x0 = 0<BR>\n" );
document.write( "210 and 330 for sin(x0 = -1/2\r<BR>\n" );
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document.write( " </SPAN>\n" );
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