document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=870024>Question 870024</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I read through Question 869777, but did not understand the response given to it, is there perhaps another way of wording it? I appreciate the answer supplied, but can't seem to wrap my head around the concept\r<BR>\n" );
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document.write( "Can you help me to solve the following question:<BR>\n" );
document.write( "The vertical displacement of the end of a robot arm (in mm) at time t (in seconds) is given by<BR>\n" );
document.write( "y=1+4cos(2t)-4sin(4t)<BR>\n" );
document.write( "a) find all times, t>0 (in exact form i.e. in terms of pi) where the vertical displacement is 1mm, i.e. y=1<BR>\n" );
document.write( "b) what is the first time, t>0 that the vertical displacement is 1mm? give you answer exactly and to 2 decimal places? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #524651 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=524651\">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=y=1%2B4cos%282t%29-4sin%284t%29=1%2B4 ALIGN=MIDDLE  ALT=\"y=1%2B4cos%282t%29-4sin%284t%29=1%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B%22 ALIGN=MIDDLE  ALT=\"%22%5B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29-sin%284t%29 ALIGN=MIDDLE  ALT=\"cos%282t%29-sin%284t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a periodic function, waving above and below <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , whose graph looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C300%2C-1.6%2C6.4%2C-7%2C9%2C1%2B4cos%282x%29-4sin%284x%29%2C1%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C300%2C-1.6%2C6.4%2C-7%2C9%2C1%2B4cos%282x%29-4sin%284x%29%2C1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Its period is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > because at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2Bpi ALIGN=MIDDLE  ALT=\"t%2Bpi\"   DISABLED_style= \"max-width:90%; height:auto;\" > the function has the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > value:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2B4cos%282t%2B2pi%29-4sin%284t%2B4pi%29=y=1%2B4cos%282t%29-4sin%284t%29 ALIGN=MIDDLE  ALT=\"y=1%2B4cos%282t%2B2pi%29-4sin%284t%2B4pi%29=y=1%2B4cos%282t%29-4sin%284t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The function will take the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > an infinite number of times.<BR>\n" );
document.write( "For an answer to part a) we can list the first few in order and then write \", ...\" or we can write it as a formula.<BR>\n" );
document.write( "Either way, it is not a simple answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The double angle trigonometric identity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282A%29=2sin%28A%29%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"sin%282A%29=2sin%28A%29%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be applied, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=2t ALIGN=MIDDLE  ALT=\"A=2t\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2B4 ALIGN=MIDDLE  ALT=\"y=1%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B%22 ALIGN=MIDDLE  ALT=\"%22%5B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29-2sin%282t%29cos%282t%29 ALIGN=MIDDLE  ALT=\"cos%282t%29-2sin%282t%29cos%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Taking out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29 ALIGN=MIDDLE  ALT=\"cos%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a common factor, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2B4 ALIGN=MIDDLE  ALT=\"y=1%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B%22 ALIGN=MIDDLE  ALT=\"%22%5B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29%281-2sin%282t%29%29 ALIGN=MIDDLE  ALT=\"cos%282t%29%281-2sin%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2B4%2Acos%282t%29%2A%281-2sin%282t%29%29 ALIGN=MIDDLE  ALT=\"y=1%2B4%2Acos%282t%29%2A%281-2sin%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When will we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "We have to solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=1%2B4%2Acos%282t%29%2A%281-2sin%282t%29%29 ALIGN=MIDDLE  ALT=\"1=1%2B4%2Acos%282t%29%2A%281-2sin%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Acos%282t%29%2A%281-2sin%282t%29%29=0 ALIGN=MIDDLE  ALT=\"4%2Acos%282t%29%2A%281-2sin%282t%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That will be true when one of those factors (either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29 ALIGN=MIDDLE  ALT=\"cos%282t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-2sin%282t%29%29 ALIGN=MIDDLE  ALT=\"%281-2sin%282t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) is zero.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282t%29=0 ALIGN=MIDDLE  ALT=\"cos%282t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t=pi%2F2 ALIGN=MIDDLE  ALT=\"2t=pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t=3pi%2F2 ALIGN=MIDDLE  ALT=\"2t=3pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or anything you can get by 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;\" > to a previous answer.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=pi%2F4 ALIGN=MIDDLE  ALT=\"t=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=3pi%2F4 ALIGN=MIDDLE  ALT=\"t=3pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or anything you can get by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to a previous answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-2sin%282t%29=0 ALIGN=MIDDLE  ALT=\"1-2sin%282t%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282t%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%282t%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t=pi%2F6 ALIGN=MIDDLE  ALT=\"2t=pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t=5pi%2F6 ALIGN=MIDDLE  ALT=\"2t=5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or anything you can get by 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;\" > to a previous answer.<BR>\n" );
document.write( "That means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=pi%2F12 ALIGN=MIDDLE  ALT=\"t=pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=5pi%2F12 ALIGN=MIDDLE  ALT=\"t=5pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or anything you can get by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to a previous answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "a) The list of times when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F12 ALIGN=MIDDLE  ALT=\"pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4=3pi%2F12 ALIGN=MIDDLE  ALT=\"pi%2F4=3pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F12 ALIGN=MIDDLE  ALT=\"5pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3pi%2F4=9pi%2F12 ALIGN=MIDDLE  ALT=\"3pi%2F4=9pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13pi%2F12=pi%2F12%2Bpi ALIGN=MIDDLE  ALT=\"13pi%2F12=pi%2F12%2Bpi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C%22 ALIGN=MIDDLE  ALT=\"%22%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22...%22 ALIGN=MIDDLE  ALT=\"%22...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and keep adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to previous answers).<BR>\n" );
document.write( "You could state that the answers are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28n%2B1%29pi%2F12 ALIGN=MIDDLE  ALT=\"t=%28n%2B1%29pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28n%2B3%29pi%2F12 ALIGN=MIDDLE  ALT=\"t=%28n%2B3%29pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28n%2B5%29pi%2F12 ALIGN=MIDDLE  ALT=\"t=%28n%2B5%29pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28n%2B9%29pi%2F12 ALIGN=MIDDLE  ALT=\"t=%28n%2B9%29pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for any non-negative integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I could not think of a simple and elegant way to express that as one formula.<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) The first time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1 ALIGN=MIDDLE  ALT=\"y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > happens when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28t=pi%2F12=0.26%29 ALIGN=MIDDLE  ALT=\"highlight%28t=pi%2F12=0.26%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );