document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=979670>Question 979670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of the equation in the interval <BR>\n" );
document.write( "[0, 2 pi)<BR>\n" );
document.write( "-sin2x+2cosx=0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #600961 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52778)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=600961\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "The equation is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2cosx ALIGN=MIDDLE  ALT=\"2cosx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2x ALIGN=MIDDLE  ALT=\"sin2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)\r<BR>\n" );
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document.write( "Apply the formula for sines of the double argument: &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2x ALIGN=MIDDLE  ALT=\"sin2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sinx%2Acosx ALIGN=MIDDLE  ALT=\"2sinx%2Acosx\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; (see, &nbsp;for example, &nbsp;the lesson &nbsp;<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument.lesson>Trigonometric functions of multiply argument</A>&nbsp; in this site). \r<BR>\n" );
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document.write( "Then the equation takes the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2cosx ALIGN=MIDDLE  ALT=\"2cosx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Acosx%2Asinx ALIGN=MIDDLE  ALT=\"2%2Acosx%2Asinx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)\r<BR>\n" );
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document.write( "One solution of this equation is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosx ALIGN=MIDDLE  ALT=\"cosx\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;which gives &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3pi%2F2 ALIGN=MIDDLE  ALT=\"3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;(regarding the interval [0, 2pi) ).\r<BR>\n" );
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document.write( "Further, &nbsp;let us assume that &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosx ALIGN=MIDDLE  ALT=\"cosx\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is not equal to zero. &nbsp;Then we can reduce the equation &nbsp;(2) dividing its both sides by &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Acosx ALIGN=MIDDLE  ALT=\"2%2Acosx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;You will get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sinx ALIGN=MIDDLE  ALT=\"sinx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "which gives the solutions &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; we just obtained above. \r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;The solutions are &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3pi%2F2 ALIGN=MIDDLE  ALT=\"3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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