document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1077821>Question 1077821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions of each of the equations in the interval [0,2pi).\r<BR>\n" );
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document.write( "a) sin(x+pi/3)+sin(x&#8722;pi/3)=1\r<BR>\n" );
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document.write( "b) tan(x+pi)+2sin(x+pi)=0\r<BR>\n" );
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document.write( "c) cos(x&#8722;pi/2)+sin2x=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #692369 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=692369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F3%29=1 ALIGN=MIDDLE  ALT=\"sin%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F3%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the trigonometric identities<BR>\n" );
document.write( "for sine of sum an difference of two angles,<BR>\n" );
document.write( "the equation can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29cos%28pi%2F3%29%2Bcos%28x%29sin%28pi%2F3%29 ALIGN=MIDDLE  ALT=\"sin%28x%29cos%28pi%2F3%29%2Bcos%28x%29sin%28pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29cos%28pi%2F3%29-cos%28x%29sin%28pi%2F3%29=1 ALIGN=MIDDLE  ALT=\"sin%28x%29cos%28pi%2F3%29-cos%28x%29sin%28pi%2F3%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Taking out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29 ALIGN=MIDDLE  ALT=\"sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29 ALIGN=MIDDLE  ALT=\"sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as common factors<BR>\n" );
document.write( "the equation can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%28sin%28pi%2F3%29-sin%28pi%2F3%29%29=1 ALIGN=MIDDLE  ALT=\"cos%28x%29%28sin%28pi%2F3%29-sin%28pi%2F3%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%280%29=1 ALIGN=MIDDLE  ALT=\"cos%28x%29%280%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29=1 ALIGN=MIDDLE  ALT=\"sin%28x%29%28cos%28pi%2F3%29%2Bcos%28pi%2F3%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F3%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28pi%2F3%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we re-write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2F2%2B1%2F2%29=1 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2F2%2B1%2F2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=1 ALIGN=MIDDLE  ALT=\"sin%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the interval [0,2pi), that happens only for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F2 ALIGN=MIDDLE  ALT=\"x=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C200%2C-0.5%2C6.5%2C-1.2%2C1.3%2Cgraph%28600%2C200%2C-0.5%2C6.5%2C-1.2%2C1.3%2Csin%28x%2Bpi%2F3%29%2Bsin%28x-pi%2F3%29%2C1%29%2Ccircle%28pi%2F2%2C1%2C0.02%29%2Clocate%281.5%2C0.97%2Cpi%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%2Bpi%29%2B2sin%28x%2Bpi%29=0 ALIGN=MIDDLE  ALT=\"tan%28x%2Bpi%29%2B2sin%28x%2Bpi%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Based on trigonometric identities, the equation can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%29-2sin%28x%29=0 ALIGN=MIDDLE  ALT=\"tan%28x%29-2sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Fcos%28x%29-2sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Fcos%28x%29-2sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, with some algebra, it can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2Fcos%28x%29-2%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2Fcos%28x%29-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281-2cos%28x%29%29%2Fcos%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%281-2cos%28x%29%29%2Fcos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The numerator is zero when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=pi%29 ALIGN=MIDDLE  ALT=\"highlight%28x=pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The numerator is also zero when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2cos%28x%29=0 ALIGN=MIDDLE  ALT=\"1-2cos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28x%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=pi%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=5pi%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=5pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For none of those values of x, is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > zero,<BR>\n" );
document.write( "so they are all valid solutions.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C200%2C-0.5%2C6.5%2C-1.2%2C1.2%2Cgraph%28600%2C200%2C-0.5%2C6.5%2C-1.2%2C1.2%2Ctan%28x%2Bpi%29%2B2sin%28x%2Bpi%29%29%2Ccircle%280%2C0%2C0.03%29%2Clocate%280.03%2C0.2%2C0%29%2C%0D%0Acircle%28pi%2C0%2C0.03%29%2Clocate%283.2%2C0.3%2Cpi%29%2C%0D%0Acircle%28pi%2F3%2C0%2C0.03%29%2Clocate%281.05%2C0%2Cpi%2F3%29%2C%0D%0Acircle%285pi%2F3%2C0%2C0.03%29%2Clocate%285.3%2C0%2C5pi%2F3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-pi%2F2%29%2Bsin%282x%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-pi%2F2%29%2Bsin%282x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Or did you mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-pi%2F2%29%2Bsin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-pi%2F2%29%2Bsin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead?)<BR>\n" );
document.write( "Using trigonometric identities,<BR>\n" );
document.write( "the equation can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28-%28x-pi%2F2%29%29%2Bsin%282x%29=0 ALIGN=MIDDLE  ALT=\"cos%28-%28x-pi%2F2%29%29%2Bsin%282x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F2-x%29%2Bsin%282x%29=0 ALIGN=MIDDLE  ALT=\"cos%28pi%2F2-x%29%2Bsin%282x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Bsin%282x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Bsin%282x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If the second term was really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression= ALIGN=MIDDLE  ALT=\"\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "using the trig identity for double angles,<BR>\n" );
document.write( "the equation can be re-written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2B2sin%28x%29cos%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%2B2sin%28x%29cos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2B2cos%28x%29%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2B2cos%28x%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2B2cos%28x%29%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2B2cos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=pi%29 ALIGN=MIDDLE  ALT=\"highlight%28x=pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2B2cos%28x%29%29 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2B2cos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also zero when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2cos%28x%29=0 ALIGN=MIDDLE  ALT=\"1%2B2cos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=-1%2F2 ALIGN=MIDDLE  ALT=\"cos%28x%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the interval [0,2pi), that happens for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=2pi%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=2pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=4pi%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=4pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C260%2C-0.5%2C6.5%2C-1.8%2C1.8%2Cgraph%28600%2C260%2C-0.5%2C6.5%2C-1.8%2C1.8%2Ccos%28x-pi%2F2%29%2Bsin%282x%29%29%2Ccircle%280%2C0%2C0.03%29%2Clocate%280.05%2C0.2%2C0%29%2C%0D%0Acircle%28pi%2C0%2C0.03%29%2Clocate%283.05%2C0.25%2Cpi%29%2C%0D%0Acircle%282pi%2F3%2C0%2C0.03%29%2Clocate%282.1%2C0.5%2C2pi%2F3%29%2C%0D%0Acircle%284pi%2F3%2C0%2C0.03%29%2Clocate%284.2%2C0.5%2C4pi%2F3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-pi%2F2%29%2Bsin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-pi%2F2%29%2Bsin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Bsin%5E2%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Bsin%5E2%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%281%2Bsin%28x%29%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29%281%2Bsin%28x%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "in the interval [0,2pi) has solutions when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0 ALIGN=MIDDLE  ALT=\"sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bsin%28x%29=0 ALIGN=MIDDLE  ALT=\"1%2Bsin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=-1 ALIGN=MIDDLE  ALT=\"sin%28x%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3pi%2F2 ALIGN=MIDDLE  ALT=\"x=3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );