document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=592892>Question 592892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the first TWO non-negative asymptotes and the first negative asymptote of the graph of y  in radians<BR>\n" );
document.write( " y=2sec(x&#8722;&#960;/2)+1 . \r<BR>\n" );
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document.write( "smallest non-negative asymptote: x=   \r<BR>\n" );
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document.write( "second non-negative asymptote: x=   \r<BR>\n" );
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document.write( "first negative asymptote: x=  <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #376125 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=376125\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We will first find a general solution for all the asymptotes. Then we will use that to find the specific ones the problem asks you to find.<br><BR>\n" );
document.write( "The (vertical) asymptotes will occur for x values that make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28x+-+pi%2F2%29 ALIGN=MIDDLE  ALT=\"sec%28x+-+pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > undefined. Since sec is the reciprocal of cos, sec is undefined when cos is zero. So we are interested in the solution to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-pi%2F2%29+=+0 ALIGN=MIDDLE  ALT=\"cos%28x-pi%2F2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You might be able to figure this out in your head. If not, we can make it easier to solve by using the cos(A-B) formula:<BR>\n" );
document.write( "cos(A-B) = cos(A)*cos(B) + sin(A)sin(B)<BR>\n" );
document.write( "With A = x and B = <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;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x-pi%2F2%29+=+0 ALIGN=MIDDLE  ALT=\"cos%28x-pi%2F2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%2Acos%28pi%2F2%29+%2B+sin%28x%29%2Asin%28pi%2F2%29+=+0 ALIGN=MIDDLE  ALT=\"cos%28x%29%2Acos%28pi%2F2%29+%2B+sin%28x%29%2Asin%28pi%2F2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cos and sin 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;\" > are known. Substituting these values in we get:<BR>\n" );
document.write( "cos(x)*0 + sin(x)*1 = 0<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "sin(x) = 0<br><BR>\n" );
document.write( "This is easily solved. Since sin(x) = 0 at 0 and at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the general solution is<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"0+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (where \"n\" is any integer)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi+%2B+2pi%2An ALIGN=MIDDLE  ALT=\"pi+%2B+2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (where \"n\" is any integer)<br><BR>\n" );
document.write( "From the general solution above, we can now find the desired specific solutions. Just play around with different integer values for \"n\" until you find the two smallest non-negative asymptotes and the first negative asymptote. (Reread the start of this solution to remind yourself why these solutions to sin(x)=0 turn out to be asymptotes for your original equation.)<br><BR>\n" );
document.write( "(You'll find that using n=0 in the first equation and n=0 in the second equation will give you the two smallest non-negative asymptotes and using n = -1 in the second equation will give you the first negative asymptote.) </SPAN>\n" );
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