document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1112429>Question 1112429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the reference angle for a 960 degree angle? show all work please  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #727473 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=727473\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is what a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > looks like,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.3%2C-1.2%2C1.2%2Cgrid%280%29%2C%0D%0Acircle%280%2C0%2C1%29%2Cred%28arc%280%2C0%2C0.6%2C0.6%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C.7%2C.6%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C.7%2C.7%2C90%2C180%29%29%2C%0D%0Ared%28arc%280%2C0%2C.8%2C.7%2C0%2C90%29%29%2C%0D%0Ared%28arc%280%2C0%2C.8%2C.8%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C.9%2C.8%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C.9%2C.9%2C90%2C180%29%29%2C%0D%0Ared%28arc%280%2C0%2C1%2C.9%2C0%2C90%29%29%2C%0D%0Ared%28arc%280%2C0%2C1%2C1%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C1.1%2C1%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C1.1%2C1.1%2C120%2C180%29%29%2C%0D%0Ared%28triangle%28-0.275%2C-0.476%2C-0.275%2C-0.426%2C-0.325%2C-0.476%29%29%2C%0D%0Alocate%28-0.7%2C-0.3%2Cred%28960%5Eo%29%29%2C%0D%0Aarrow%280%2C0%2C-0.6%2C-1.039%29%2Carrow%280%2C0%2C1.25%2C0%29%2C%0D%0Alocate%280.5%2C1.1%2CQUADRANT%29%2Clocate%281.05%2C1.1%2CI%29%2C%0D%0Alocate%28-1%2C1.1%2CQUADRANT%29%2Clocate%28-.45%2C1.1%2CII%29%2C%0D%0Alocate%280.5%2C-1.04%2CQUADRANT%29%2Clocate%281.05%2C-1.04%2CIV%29%2C%0D%0Alocate%28-1.05%2C-1.04%2CQUADRANT%29%2Clocate%28-.5%2C-1.04%2CIII%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > It's 2 turns and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo ALIGN=MIDDLE  ALT=\"240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (all counterclockwise, of course).<BR>\n" );
document.write( "When you divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > the quotient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the remainder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo ALIGN=MIDDLE  ALT=\"240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You do not need to divide (unless the angle measure is huge).<BR>\n" );
document.write( "You can keep subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (one turn) until you get less than one turn.<BR>\n" );
document.write( "You can write that as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo-360%5Eo-360%5Eo=240%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo-360%5Eo-360%5Eo=240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >  , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo-2%28360%5Eo%29=240%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo-2%28360%5Eo%29=240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo-720%5Eo=240%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo-720%5Eo=240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now you have to work with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo ALIGN=MIDDLE  ALT=\"240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the co-terminal angle of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=960%5Eo ALIGN=MIDDLE  ALT=\"960%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "An angle measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo ALIGN=MIDDLE  ALT=\"240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the third quadrant,<BR>\n" );
document.write( "where all the angles between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=270%5Eo ALIGN=MIDDLE  ALT=\"270%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > are located.<BR>\n" );
document.write( "You can write that as <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo%3C240%5Eo%3C270%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo%3C240%5Eo%3C270%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The reference angle is a symmetrical angle in quadrant I, and that angle has the same absolute value for all trigonometric functions.<BR>\n" );
document.write( "For quadrant III, it is easy, you just subtract half a turn<, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that gives you the angle with the opposite ray for a terminal side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.3%2C-1.2%2C1.2%2Cgrid%280%29%2C%0D%0Acircle%280%2C0%2C1%29%2Cred%28arc%280%2C0%2C0.6%2C0.6%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C.7%2C.6%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C.7%2C.7%2C90%2C180%29%29%2C%0D%0Ared%28arc%280%2C0%2C.8%2C.7%2C0%2C90%29%29%2C%0D%0Ared%28arc%280%2C0%2C.8%2C.8%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C.9%2C.8%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C.9%2C.9%2C90%2C180%29%29%2C%0D%0Ared%28arc%280%2C0%2C1%2C.9%2C0%2C90%29%29%2C%0D%0Ared%28arc%280%2C0%2C1%2C1%2C270%2C360%29%29%2C%0D%0Ared%28arc%280%2C0%2C1.1%2C1%2C180%2C270%29%29%2C%0D%0Ared%28arc%280%2C0%2C1.1%2C1.1%2C120%2C180%29%29%2C%0D%0Ared%28triangle%28-0.275%2C-0.476%2C-0.275%2C-0.426%2C-0.325%2C-0.476%29%29%2C%0D%0Alocate%28-0.7%2C-0.3%2Cred%28960%5Eo%29%29%2C%0D%0Aarrow%280%2C0%2C-0.6%2C-1.039%29%2Carrow%280%2C0%2C1.25%2C0%29%2C%0D%0Alocate%280.5%2C1.2%2CQUADRANT%29%2Clocate%281.05%2C1.2%2CI%29%2C%0D%0Alocate%28-1%2C1.1%2CQUADRANT%29%2Clocate%28-.45%2C1.1%2CII%29%2C%0D%0Alocate%280.5%2C-1.04%2CQUADRANT%29%2Clocate%281.05%2C-1.04%2CIV%29%2C%0D%0Alocate%28-1.05%2C-1.04%2CQUADRANT%29%2Clocate%28-.5%2C-1.04%2CIII%29%2C%0D%0Aarrow%280%2C0%2C0.6%2C1.039%29%2Cgreen%28arc%280%2C0%2C1.2%2C1.2%2C-60%2C360%29%29%2C%0D%0Agreen%28triangle%280.3%2C0.52%2C0.3%2C0.47%2C0.35%2C0.52%29%29%2C%0D%0Alocate%280.4%2C0.6%2Cgreen%28240%5Eo-180%5Eo=60%5Eo%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The reference angle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo-180%5Eo=highlight%2860%5Eo%29 ALIGN=MIDDLE  ALT=\"240%5Eo-180%5Eo=highlight%2860%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );