document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=768031>Question 768031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having real trouble grasping the concepts.<BR>\n" );
document.write( "Topic: Solving Trigonometric Equations<BR>\n" );
document.write( "Q:<BR>\n" );
document.write( "Find all solutions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sin%28+2x+%29+=+-sqrt%28+3+%29%2F2+ ALIGN=MIDDLE  ALT=\"+sin%28+2x+%29+=+-sqrt%28+3+%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the interval [0,2pi)\r<BR>\n" );
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document.write( "Answer Choices:\r<BR>\n" );
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document.write( "1. None of these\r<BR>\n" );
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document.write( "2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+pi%2F3+ ALIGN=MIDDLE  ALT=\"+x+=+pi%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284%2Api%29%2F3+ ALIGN=MIDDLE  ALT=\"+%284%2Api%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%285%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%285%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%287%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%287%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%282%2Api%29%2F3+ ALIGN=MIDDLE  ALT=\"+x+=+%282%2Api%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%285%2Api%29%2F3+ ALIGN=MIDDLE  ALT=\"+%285%2Api%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%285%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%285%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2811%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%2811%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "4. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+pi%2F3+ ALIGN=MIDDLE  ALT=\"+x+=+pi%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%285%2Api%29%2F3+ ALIGN=MIDDLE  ALT=\"+%285%2Api%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <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;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%2811%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%2811%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "5. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%282pi%29%2F3+ ALIGN=MIDDLE  ALT=\"+x+=+%282pi%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284%2Api%29%2F3+ ALIGN=MIDDLE  ALT=\"+%284%2Api%29%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <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;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%287%2Api%29%2F6+ ALIGN=MIDDLE  ALT=\"+%287%2Api%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Please note: I am only showing ans choices as a frame of reference; I'm more interested in a detailed review of how to work the problem. Also, do I need to draw a unit circle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #468014 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=468014\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Find the two points on the unit circle illustrated where the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\frac{\sqrt{3}}{2}\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://prec.alcul.us/wp-content/uploads/unitcircle.gif\">\r<BR>\n" );
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document.write( "Now, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> has to be in the interval [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0,2\pi\">), but that means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x\"> is in the interval [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0,4\pi\">).   Hence the two points you found on the circle are actually 4 possibilities for the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x\">, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{4\pi}{3},\,\frac{5\pi}{3},\,\frac{10\pi}{3},\,\frac{11\pi}{3}\">.  Divide each possibility for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x\"> by 2, and you will have 4 numbers that match one of your answer choices.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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