document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=732671>Question 732671</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions to the equation. \r<BR>\n" );
document.write( "\n" );
document.write( "cos x = -1/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #448016 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=448016\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2860%5Eo%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%2860%5Eo%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <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;\" > if measuring angles in radians).<BR>\n" );
document.write( "Negative cosines happen only in the second and third quadrants.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28180%5Eo-60%5Eo%29=cos%28120%5Eo%29+=-cos%2860%5Eo%29=-1%2F2 ALIGN=MIDDLE  ALT=\"cos%28180%5Eo-60%5Eo%29=cos%28120%5Eo%29+=-cos%2860%5Eo%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi-pi%2F3%29=cos%282pi%2F3%29=-cos%28pi%2F3%29=-1%2F2 ALIGN=MIDDLE  ALT=\"cos%28pi-pi%2F3%29=cos%282pi%2F3%29=-cos%28pi%2F3%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in radians) shows us one of the solutions.<BR>\n" );
document.write( "An angle with the same cosine in the third quadrant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-120%5Eo ALIGN=MIDDLE  ALT=\"-120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2pi%2F3 ALIGN=MIDDLE  ALT=\"-2pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > in radians and that is another solution.<BR>\n" );
document.write( "The other solutions are all angles coterminal with those two angles.<BR>\n" );
document.write( "They can all be expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo+%2B-+k%2A120%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo+%2B-+k%2A120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi+%2B-+2k%2Api%2F3 ALIGN=MIDDLE  ALT=\"2pi+%2B-+2k%2Api%2F3\"   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;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );