document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252605>Question 252605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The height of the ocean at the dock is modeled by the function, h= 3sin(pi*t/4) +5 where h is measured in feet and t is the time in hours. If t=0 refers to 12:00 a.m., what is the height of the ocean at 10:00 a.m.?\r<BR>\n" );
document.write( "\n" );
document.write( "find the exact value of the vertical asymptotes for 0<=x<=pi for the function y=cot(3x)\r<BR>\n" );
document.write( "\n" );
document.write( "if sin(x)= -3/5 with x in the quadrant 4, find the sec(x)\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks so much!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184615 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184615\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to get you going.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "# 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since \"t=0 refers to 12:00 a.m\", this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10 ALIGN=MIDDLE  ALT=\"t=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > refers to 10:00 a.m. (just add 10 hours to 0)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%28pi%2At%2F4%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%28pi%2At%2F4%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%28pi%2A10%2F4%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%28pi%2A10%2F4%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10 ALIGN=MIDDLE  ALT=\"t=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%28pi%2A5%2F2%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%28pi%2A5%2F2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%285pi%2F2%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%285pi%2F2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%285pi%2F2-2pi%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%285pi%2F2-2pi%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the argument (this is valid because you'll end up on a coterminal angle).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2Asin%28pi%2F2%29%2B5 ALIGN=MIDDLE  ALT=\"h=3%2Asin%28pi%2F2%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2A1%2B5 ALIGN=MIDDLE  ALT=\"h=3%2A1%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the unit circle to evaluate the sine of <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;\" > to get 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=3%2B5 ALIGN=MIDDLE  ALT=\"h=3%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=8 ALIGN=MIDDLE  ALT=\"h=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the height of the ocean at 10:00 a.m. is 8 feet. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "# 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Remember that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%28x%29=1%2Ftan%28x%29 ALIGN=MIDDLE  ALT=\"cot%28x%29=1%2Ftan%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28x%29=sin%28x%29%2Fcos%28x%29 ALIGN=MIDDLE  ALT=\"tan%28x%29=sin%28x%29%2Fcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%283x%29=1%2Ftan%283x%29=1%2F%28sin%283x%29%2Fcos%283x%29%29=cos%283x%29%2Fsin%283x%29 ALIGN=MIDDLE  ALT=\"cot%283x%29=1%2Ftan%283x%29=1%2F%28sin%283x%29%2Fcos%283x%29%29=cos%283x%29%2Fsin%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or in short, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%283x%29=cos%283x%29%2Fsin%283x%29 ALIGN=MIDDLE  ALT=\"cot%283x%29=cos%283x%29%2Fsin%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To find the vertical asymptotes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%283x%29=cos%283x%29%2Fsin%283x%29 ALIGN=MIDDLE  ALT=\"cot%283x%29=cos%283x%29%2Fsin%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we'll set the denominator equal to zero and solve for 'x' (since division by zero is undefined).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%283x%29=0 ALIGN=MIDDLE  ALT=\"sin%283x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=arcsin%280%29 ALIGN=MIDDLE  ALT=\"3x=arcsin%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the arcsine of both sides.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B2pi%2An ALIGN=MIDDLE  ALT=\"x=0%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2B2pi%2An ALIGN=MIDDLE  ALT=\"x=pi%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate the arcsine of 0 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Don't forget to add on multiples 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;\" > to each solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%280%2B2pi%2An%29%2F3 ALIGN=MIDDLE  ALT=\"x=%280%2B2pi%2An%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28pi%2B2pi%2An%29%2F3 ALIGN=MIDDLE  ALT=\"x=%28pi%2B2pi%2An%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 3 to isolate 'x' in each case. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As a shortcut, you can condense the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29pi%2An ALIGN=MIDDLE  ALT=\"x=%281%2F3%29pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'n' is an integer. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29pi%2An ALIGN=MIDDLE  ALT=\"x=%281%2F3%29pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where 'n' is an integer, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%283x%29=0 ALIGN=MIDDLE  ALT=\"sin%283x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=pi ALIGN=MIDDLE  ALT=\"0%3C=x%3C=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that we're only going to look at the solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for n=0), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F3 ALIGN=MIDDLE  ALT=\"x=pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (where n=1), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2pi%2F3 ALIGN=MIDDLE  ALT=\"x=2pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (when n=2), and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > (when x=3). Note: any other solution is outside the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=pi ALIGN=MIDDLE  ALT=\"0%3C=x%3C=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the four vertical asymptotes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%283x%29 ALIGN=MIDDLE  ALT=\"cot%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi%2F3 ALIGN=MIDDLE  ALT=\"x=pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2pi%2F3 ALIGN=MIDDLE  ALT=\"x=2pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=pi ALIGN=MIDDLE  ALT=\"x=pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );