document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=871008>Question 871008</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the solutions in the interval [0, 2pi)\r<BR>\n" );
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document.write( "2 cos 3x = 1\r<BR>\n" );
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document.write( "This is due tomorrow and I am very confused, any and all help is greatly appreciated :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #525258 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=525258\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2cos%283x%29=1 ALIGN=MIDDLE  ALT=\"2cos%283x%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=cos%283x%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%283x%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cosine function takes all the positive values between 0 and 1, without repeating in quadrant I and again in quadrant IV.<BR>\n" );
document.write( "In quadrants II and III cosine has negative values.<BR>\n" );
document.write( "In quadrant I (between 0 and <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;\" > , or if you prefer between<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5Eo ALIGN=MIDDLE  ALT=\"0%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "we have <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;\" > .<BR>\n" );
document.write( "In quadrant IV , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-pi%2F3 ALIGN=MIDDLE  ALT=\"-pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a cosine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
document.write( "Since cosine has a period 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;\" > , adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > you get an angle with the same cosine.<BR>\n" );
document.write( "So all the angles with a cosine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k%2Api+%2B-+pi%2F3 ALIGN=MIDDLE  ALT=\"2k%2Api+%2B-+pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286k+%2B-+1%29pi%2F3 ALIGN=MIDDLE  ALT=\"%286k+%2B-+1%29pi%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;\" ><BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%283x%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%283x%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=%286k+%2B-+1%29pi%2F3 ALIGN=MIDDLE  ALT=\"3x=%286k+%2B-+1%29pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%286k+%2B-+1%29pi%2F9 ALIGN=MIDDLE  ALT=\"x=%286k+%2B-+1%29pi%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A0+-+1%29pi%2F9=-pi%2F9%3C0 ALIGN=MIDDLE  ALT=\"%286%2A0+-+1%29pi%2F9=-pi%2F9%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A3%2B1%29pi%2F9=highlight%2819pi%2F9%3E2pi%29 ALIGN=MIDDLE  ALT=\"%286%2A3%2B1%29pi%2F9=highlight%2819pi%2F9%3E2pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are outside the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+0+%2C%22 ALIGN=MIDDLE  ALT=\"%22%5B+0+%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The solutions in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B+0+%2C%22 ALIGN=MIDDLE  ALT=\"%22%5B+0+%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A0+%2B+1%29pi%2F9=highlight%28pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A0+%2B+1%29pi%2F9=highlight%28pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A1+-+1%29pi%2F9=highlight%285pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A1+-+1%29pi%2F9=highlight%285pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A1+%2B+1%29pi%2F9=highlight%287pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A1+%2B+1%29pi%2F9=highlight%287pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A2+-+1%29pi%2F9=highlight%2811pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A2+-+1%29pi%2F9=highlight%2811pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A2%2B1%29pi%2F9=highlight%2813pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A2%2B1%29pi%2F9=highlight%2813pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A3+-+1%29pi%2F9=highlight%2817pi%2F9%29 ALIGN=MIDDLE  ALT=\"%286%2A3+-+1%29pi%2F9=highlight%2817pi%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .  </SPAN>\n" );
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