document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=870286>Question 870286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation!<BR>\n" );
document.write( "(a)Solve approximately the equations:<BR>\n" );
document.write( "(i)2 sin x + cos x = 1.5<BR>\n" );
document.write( "(ii)2 sin x + cos x = 0\r<BR>\n" );
document.write( "\n" );
document.write( "(b)(i)3 cos x - 4 sin x + 1 = 0<BR>\n" );
document.write( "(ii) 3 cos x = 4 sin x<BR>\n" );
document.write( "PLEASE REPLY ASAP! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #524915 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=524915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I am learning something important from this problem (and from the wiki I found once I figured out how to ask the question).<BR>\n" );
document.write( "Two of your problems (aii and bii) were easy to answer without much algebra and/or trigonometry.<BR>\n" );
document.write( "The other two were clearly related, but required more algebra work.<BR>\n" );
document.write( "There had to be a common strategy to solve all four problems,<BR>\n" );
document.write( "and using trigonometric identities had to be the key,<BR>\n" );
document.write( "but I could not see how to use trigonometric identities to get where I wanted to get.<BR>\n" );
document.write( "A quick internet search led me to the common strategy to solve the problems.<BR>\n" );
document.write( " <BR>\n" );
document.write( "EACH PROBLEM AS A SEPARATE STRUGGLE:<BR>\n" );
document.write( "(a)(ii) <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=0 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29=-cos%28x%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29=-cos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29%2Fcos%28x%29=-1%2F2 ALIGN=MIDDLE  ALT=\"sin%28x%29%2Fcos%28x%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%29=-1%2F2 ALIGN=MIDDLE  ALT=\"tan%28x%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From there we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.43648 ALIGN=MIDDLE  ALT=\"x=-0.43648\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26.5651%5Eo ALIGN=MIDDLE  ALT=\"-26.5651%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > in degrees) is an answer.<BR>\n" );
document.write( "Of course we know that there are infinite answers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians) apart from each other,<BR>\n" );
document.write( "because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=tan%28x%29 ALIGN=MIDDLE  ALT=\"y=tan%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a periodic function with a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > expressed in radians).\r<BR>\n" );
document.write( "\n" );
document.write( "We could say that all of our approximate answers can be expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=k%2Api-0.43648%29 ALIGN=MIDDLE  ALT=\"highlight%28x=k%2Api-0.43648%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28K=integer%29 ALIGN=MIDDLE  ALT=\"highlight%28K=integer%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > measured in radians).<BR>\n" );
document.write( "The same strategy can be used to solve (b)(ii).<BR>\n" );
document.write( "(I found <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=2.6779%2Bk%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=2.6779%2Bk%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a general solution).<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a)(i)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=1.5-2sin%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29=1.5-2sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29%29%5E2=2.25-6sin%28x%29%2B%282sin%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28cos%28x%29%29%5E2=2.25-6sin%28x%29%2B%282sin%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28sin%28x%29%29%5E2=2.25-6sin%28x%29%2B4%28sin%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"1-%28sin%28x%29%29%5E2=2.25-6sin%28x%29%2B4%28sin%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Calling <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29 ALIGN=MIDDLE  ALT=\"y=sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can re-write the equation above as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-y%5E2=2.25-6y%2B4y%5E2 ALIGN=MIDDLE  ALT=\"1-y%5E2=2.25-6y%2B4y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2-6y%2B1.25=0 ALIGN=MIDDLE  ALT=\"5y%5E2-6y%2B1.25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the quadratic formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-%28-6%29+%2B-+sqrt%28%28-6%29%5E2-4%2A5%2A1.25%29%29%2F%282%2A5%29=%286+%2B-+sqrt%2836-25%29%29%2F10=%286+%2B-+sqrt%2811%29%29%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate solutions are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%286%2Bsqrt%2811%29%29%2F10=0.9316625 ALIGN=MIDDLE  ALT=\"y=%286%2Bsqrt%2811%29%29%2F10=0.9316625\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%286-sqrt%2811%29%29%2F10=0.2683375 ALIGN=MIDDLE  ALT=\"y=%286-sqrt%2811%29%29%2F10=0.2683375\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With those values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29 ALIGN=MIDDLE  ALT=\"y=sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we set out to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29=0.9316625 ALIGN=MIDDLE  ALT=\"y=sin%28x%29=0.9316625\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "There are two values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > between 0 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or 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=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > that yield <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29=0.9316625 ALIGN=MIDDLE  ALT=\"y=sin%28x%29=0.9316625\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "One is in quadrant I and the other in quadrant II.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29=2%2A0.9316625=1.863325%3E1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29=2%2A0.9316625=1.863325%3E1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%3C0 ALIGN=MIDDLE  ALT=\"cos%28x%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we are looking foir a quadrant II solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=1.9426%29 ALIGN=MIDDLE  ALT=\"highlight%28x=1.9426%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28111.3%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%28111.3%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in degress) is a solution.<BR>\n" );
document.write( "Another possible value is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1.1990 ALIGN=MIDDLE  ALT=\"x=1.1990\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=68.7%5Eo ALIGN=MIDDLE  ALT=\"68.7%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >) ),<BR>\n" );
document.write( "but that is in quadrant I, with a positive <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%281.1990%29=0.03633 ALIGN=MIDDLE  ALT=\"cos%281.1990%29=0.03633\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%281.199%29%2Bcos%281.199%29=2%2A0.9316%2B0.03633=1.86332%2B0.03633=2.22665 ALIGN=MIDDLE  ALT=\"2sin%281.199%29%2Bcos%281.199%29=2%2A0.9316%2B0.03633=1.86332%2B0.03633=2.22665\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "does not satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=sin%28x%29=0.2683375 ALIGN=MIDDLE  ALT=\"y=sin%28x%29=0.2683375\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "The solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0.27167%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0.27167%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815.57%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%2815.57%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in degrees), in quadrant I, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%3E0 ALIGN=MIDDLE  ALT=\"cos%28x%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > looks promising,<BR>\n" );
document.write( "as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=2%2A0.2683375%2Bcos%28x%29=0.572675%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=2%2A0.2683375%2Bcos%28x%29=0.572675%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5 ALIGN=MIDDLE  ALT=\"1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In fact, it can be verified to satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=+1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=+1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=2%2A0.2683375%2Bcos%28x%29=0.572675%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=2%2A0.2683375%2Bcos%28x%29=0.572675%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the other hand, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.9 ALIGN=MIDDLE  ALT=\"x=2.9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , in quadrant II, while it also yields <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=0.268338 ALIGN=MIDDLE  ALT=\"sin%28x%29=0.268338\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "it yields <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29=-0.97%3C0 ALIGN=MIDDLE  ALT=\"cos%28x%29=-0.97%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and does not satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The solutions highlighted above, are the solutions between 0 and <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( "and in general, all solutions can be expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0.27167%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0.27167%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=1.9426%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=1.9426%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28K=integer%29 ALIGN=MIDDLE  ALT=\"highlight%28K=integer%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > measured in radians).<BR>\n" );
document.write( "The same strategy can be used to solve (b)(i).<BR>\n" );
document.write( "(The general solutions I found are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0.8449%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0.8449%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=3.5837%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=3.5837%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( " <BR>\n" );
document.write( "A COMMON STRATEGY:<BR>\n" );
document.write( "The left side of the equation in (a)(ii) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2sin%28x%29%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"y=2sin%28x%29%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It is a \"linear combination of sine and cosine functions\".<BR>\n" );
document.write( "That is a periodic function, like sine and cosine.<BR>\n" );
document.write( "I can see that its period is <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( "It must be possible to express it as a single trigonometric function,<BR>\n" );
document.write( "maybe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Acos%28x-D%29 ALIGN=MIDDLE  ALT=\"y=C%2Acos%28x-D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,with two constants <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where the cosine function is shifted right by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and dilated vertically by a factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now, how could I use trigonometric identities to transform<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Acos%28x%29-4%2Asin%28x%29 ALIGN=MIDDLE  ALT=\"y=3%2Acos%28x%29-4%2Asin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2sin%28x%29%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"y=2sin%28x%29%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "into a function like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Acos%28x-D%29 ALIGN=MIDDLE  ALT=\"y=C%2Acos%28x-D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "It required a lot of thinking, and on the Sunday morning after such a Saturday night, I did not trust my brain that much.<BR>\n" );
document.write( "I just googled \"linear combinations of sine and cosine functions\",<BR>\n" );
document.write( "and helped myself to someone else's thinking.<BR>\n" );
document.write( "Trigonometric identities tell us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-D%29=cos%28D%29%2Acos%28x%29%2Bsin%28D%29%2Asin%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x-D%29=cos%28D%29%2Acos%28x%29%2Bsin%28D%29%2Asin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Acos%28x-D%29=C%2A%28cos%28D%29%2Acos%28x%29%2Bsin%28D%29%2Asin%28x%29%29=Ccos%28D%29%2Acos%28x%29%2BCsin%28D%29%2Asin%28x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So if a linear combination of sine and cosine functions,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=A%2Acos%28x%29%2BB%2Asin%28x%29 ALIGN=MIDDLE  ALT=\"y=A%2Acos%28x%29%2BB%2Asin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Acos%28x-D%29 ALIGN=MIDDLE  ALT=\"y=C%2Acos%28x-D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ccos%28D%29%2Acos%28x%29%2BCsin%28D%29%2Asin%28x%29=A%2Acos%28x%29%2BB%2Asin%28x%29 ALIGN=MIDDLE  ALT=\"Ccos%28D%29%2Acos%28x%29%2BCsin%28D%29%2Asin%28x%29=A%2Acos%28x%29%2BB%2Asin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=Ccos%28D%29%2CB=CsinD%29 ALIGN=MIDDLE  ALT=\"system%28A=Ccos%28D%29%2CB=CsinD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28tan%28D%29=B%2FA%2CC%5E2=A%5E2%2BB%5E2%29 ALIGN=MIDDLE  ALT=\"system%28tan%28D%29=B%2FA%2CC%5E2=A%5E2%2BB%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Although that gives you two choices for C,<BR>\n" );
document.write( "it is a formula-driven, apparently less cumbersome, common strategy to solve all four problems.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Applying those formulas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=1%2CB=2%29 ALIGN=MIDDLE  ALT=\"system%28A=1%2CB=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28tan%28D%29=2%2CC%5E2=5%29 ALIGN=MIDDLE  ALT=\"system%28tan%28D%29=2%2CC%5E2=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle in quadrant I that has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28D%29=2 ALIGN=MIDDLE  ALT=\"tan%28D%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "measures approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=1.10715 ALIGN=MIDDLE  ALT=\"D=1.10715\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians).<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=sqrt%285%29 ALIGN=MIDDLE  ALT=\"C=sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=1.10715 ALIGN=MIDDLE  ALT=\"D=1.10715\"   DISABLED_style= \"max-width:90%; height:auto;\" > we would conclude that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=sqrt%285%29%2Acos%28x-1.10715%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=sqrt%285%29%2Acos%28x-1.10715%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We re-write the equations that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve:<BR>\n" );
document.write( "(a)(i) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=1.5 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2Acos%28x-1.10715%29=1.5 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2Acos%28x-1.10715%29=1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-1.10715%29=1.5%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%29=1.5%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-1.10715%29=0.3sqrt%285%29 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%29=0.3sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-1.10715%29=0.3sqrt%285%29 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%29=0.3sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-1.10715=0.83548%2B2k%2Api%2Cx-1.10715=-0.83548%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"system%28x-1.10715=0.83548%2B2k%2Api%2Cx-1.10715=-0.83548%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0.83548%2B1.10715%2B2k%2Api%2Cx-1.10715=-0.83548%2B1.10715%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"system%28x=0.83548%2B1.10715%2B2k%2Api%2Cx-1.10715=-0.83548%2B1.10715%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x=1.94263%2B2k%2Api%2Cx-1.10715=0.27167%2B2k%2Api%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x=1.94263%2B2k%2Api%2Cx-1.10715=0.27167%2B2k%2Api%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(a)(ii)<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Bcos%28x%29=0 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Bcos%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2Acos%28x-1.10715%29=0 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2Acos%28x-1.10715%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-1.10715%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-1.10715%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%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=cos%28x-1.10715%29=0 ALIGN=MIDDLE  ALT=\"cos%28x-1.10715%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-1.10715=pi%2F2%2B2k%2Api%2Cx-1.10715=-pi%2F2%2B2k%2Api%29 ALIGN=MIDDLE  ALT=\"system%28x-1.10715=pi%2F2%2B2k%2Api%2Cx-1.10715=-pi%2F2%2B2k%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=1.10715%2Bk%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=1.10715%2Bk%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29-4sin%28x%29 ALIGN=MIDDLE  ALT=\"3cos%28x%29-4sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=3%2CB=-4%29 ALIGN=MIDDLE  ALT=\"system%28A=3%2CB=-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28tan%28D%29=-4%2F3%2CC%5E2=3%5E2%2B4%5E2=9%2B16=25=5%5E2%29 ALIGN=MIDDLE  ALT=\"system%28tan%28D%29=-4%2F3%2CC%5E2=3%5E2%2B4%5E2=9%2B16=25=5%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle in quadrant IV that has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28D%29=-4%2F3+is ALIGN=MIDDLE  ALT=\"tan%28D%29=-4%2F3+is\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "measures approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=-0.92730 ALIGN=MIDDLE  ALT=\"D=-0.92730\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in radians).<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=5 ALIGN=MIDDLE  ALT=\"C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=-0.92730 ALIGN=MIDDLE  ALT=\"D=-0.92730\"   DISABLED_style= \"max-width:90%; height:auto;\" > we would conclude that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29-4sin%28x%29=5cos%28x%2B0.92730%29 ALIGN=MIDDLE  ALT=\"3cos%28x%29-4sin%28x%29=5cos%28x%2B0.92730%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We re-write the equations that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29-4sin%28x%29 ALIGN=MIDDLE  ALT=\"3cos%28x%29-4sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve:<BR>\n" );
document.write( "(b)(i) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29-4sin%28x%29%2B1=0 ALIGN=MIDDLE  ALT=\"3cos%28x%29-4sin%28x%29%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%28x%2B0.92730%29%2B1=0 ALIGN=MIDDLE  ALT=\"5cos%28x%2B0.92730%29%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%28x%2B0.92730%29=-1 ALIGN=MIDDLE  ALT=\"5cos%28x%2B0.92730%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2B0.92730%29=-1%2F5 ALIGN=MIDDLE  ALT=\"cos%28x%2B0.92730%29=-1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2B0.92730%29=-0.2 ALIGN=MIDDLE  ALT=\"cos%28x%2B0.92730%29=-0.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B0.92730=2k%2Api+%2B-+1.77215 ALIGN=MIDDLE  ALT=\"x%2B0.92730=2k%2Api+%2B-+1.77215\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.92730+%2B-+1.77215%2B2kpi ALIGN=MIDDLE  ALT=\"x=-0.92730+%2B-+1.77215%2B2kpi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0.8449%2B2kpi%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0.8449%2B2kpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-26994%2B2kpi ALIGN=MIDDLE  ALT=\"x=-26994%2B2kpi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which can be written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2pi-26994%2B2kpi=highlight%283.5837%2B2kpi%29 ALIGN=MIDDLE  ALT=\"x=2pi-26994%2B2kpi=highlight%283.5837%2B2kpi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(b)(ii) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29=4sin%28x%29 ALIGN=MIDDLE  ALT=\"3cos%28x%29=4sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3cos%28x%29-4sin%28x%29=0 ALIGN=MIDDLE  ALT=\"3cos%28x%29-4sin%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5cos%28x%2B0.92730%29=0 ALIGN=MIDDLE  ALT=\"5cos%28x%2B0.92730%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%2B0.92730%29=0 ALIGN=MIDDLE  ALT=\"cos%28x%2B0.92730%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B0.92730=2k%2Api+%2B-+pi%2F2 ALIGN=MIDDLE  ALT=\"x%2B0.92730=2k%2Api+%2B-+pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F2-0.9273%2Bk%2Api%29 ALIGN=MIDDLE  ALT=\"x=pi%2F2-0.9273%2Bk%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=0.6435%2Bk%2Api%29 ALIGN=MIDDLE  ALT=\"highlight%28x=0.6435%2Bk%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );