document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=873027>Question 873027</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the multi-angle below\r<BR>\n" );
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document.write( "sin(2x)= - square root of 2 / 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #526638 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=526638\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282x%29=+-sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%282x%29=+-sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > It could be that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=-pi%2F4 ALIGN=MIDDLE  ALT=\"2x=-pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=-3pi%2F4 ALIGN=MIDDLE  ALT=\"2x=-3pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for angles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AOP ALIGN=MIDDLE  ALT=\"AOP\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AOQ ALIGN=MIDDLE  ALT=\"AOQ\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-1.2%2C1.2%2C%0D%0Agreen%28triangle%280%2C0%2C-0.707%2C0%2C-0.707%2C-0.707%29%29%2C%0D%0Agreen%28rectangle%28-0.707%2C0%2C-0.657%2C-0.05%29%29%2C%0D%0Agrid%280%29%2Ccircle%280%2C0%2C1%29%2Clocate%28-0.69%2C-0.15%2Cgreen%28sqrt%282%29%2F2%29%29%2C%0D%0Agreen%28arrow%280%2C0%2C1%2C-1%29%29%2Cgreen%28arc%280%2C0%2C1%2C1%2C0%2C45%29%29%2C%0D%0Agreen%28arc%280%2C0%2C0.9%2C0.9%2C45%2C90%29%29%2Clocate%280.25%2C0%2Cgreen%28pi%2F4%29%29%2C%0D%0Agreen%28arrow%280%2C0%2C-1%2C-1%29%29%2Cgreen%28arc%280%2C0%2C0.8%2C0.8%2C90%2C135%29%29%2C%0D%0Agreen%28arc%280%2C0%2C0.9%2C0.9%2C135%2C180%29%29%2Clocate%28-0.35%2C0%2Cgreen%28pi%2F4%29%29%2C%0D%0Ablue%28arc%280%2C0%2C1.6%2C1.6%2C0%2C135%29%29%2Clocate%28-0.2%2C-0.5%2Cblue%283pi%2F4%29%29%2C%0D%0Alocate%280.02%2C0.13%2CO%29locate%281.01%2C0.12%2CA%29%2C%0D%0Alocate%280.75%2C-0.65%2CP%29%2Clocate%28-0.8%2C-0.65%2CQ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > could also be any other angle differing from those by a multiple of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Starting from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=-pi%2F2+%2B-+pi%2F4 ALIGN=MIDDLE  ALT=\"2x=-pi%2F2+%2B-+pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x=-pi%2F4%2C%22or%22%2C2x=-3pi%2F4%29 ALIGN=MIDDLE  ALT=\"system%282x=-pi%2F4%2C%22or%22%2C2x=-3pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=3pi%2F2+%2B-+pi%2F4 ALIGN=MIDDLE  ALT=\"2x=3pi%2F2+%2B-+pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x=5pi%2F4%2C%22or%22%2C2x=7pi%2F4%29 ALIGN=MIDDLE  ALT=\"system%282x=5pi%2F4%2C%22or%22%2C2x=7pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a way of expressing all those angle with one formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=%282k%2Api-pi%2F2%29+%2B-+pi%2F4=8k%2Api%2F4-2pi%2F4+%2B-+pi%2F4=%288k-2+%2B-+1%29pi%2F4 ALIGN=MIDDLE  ALT=\"2x=%282k%2Api-pi%2F2%29+%2B-+pi%2F4=8k%2Api%2F4-2pi%2F4+%2B-+pi%2F4=%288k-2+%2B-+1%29pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x=%288k-2+%2B-+1%29pi%2F4 ALIGN=MIDDLE  ALT=\"2x=%288k-2+%2B-+1%29pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=%288k-2+%2B-+1%29pi%2F8%29 ALIGN=MIDDLE  ALT=\"highlight%28x=%288k-2+%2B-+1%29pi%2F8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That gives you infinite pairs of co-terminal angles:<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-11pi%2F8 ALIGN=MIDDLE  ALT=\"x=-11pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9pi%2F8 ALIGN=MIDDLE  ALT=\"x=-9pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3pi%2F8 ALIGN=MIDDLE  ALT=\"x=-3pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-pi%2F8 ALIGN=MIDDLE  ALT=\"x=-pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1 ALIGN=MIDDLE  ALT=\"k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5pi%2F8 ALIGN=MIDDLE  ALT=\"x=5pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7pi%2F8 ALIGN=MIDDLE  ALT=\"x=7pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2 ALIGN=MIDDLE  ALT=\"k=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=13pi%2F8 ALIGN=MIDDLE  ALT=\"x=13pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15pi%2F8 ALIGN=MIDDLE  ALT=\"x=15pi%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are infinitely more pairs. </SPAN>\n" );
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