document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=821433>Question 821433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of the function\r<BR>\n" );
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document.write( "tan(beta/2), given tan(beta)= -((sqrt 5)/2), with 90 degrees < beta < 180 degrees </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #494227 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=494227\">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=90+%3C+beta+%3C+180+ ALIGN=MIDDLE  ALT=\"90+%3C+beta+%3C+180+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" > terminates in the second quadrant, where x-coordinates are negative and y-coordinates are positive.<br><BR>\n" );
document.write( "At this point a diagram may be helpful:<ul><li>Draw an angle in standard position which terminates in the 2nd quadrant. This angle will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >.</li><li>From somewhere on the terminal side draw a perpendicular down to the x-axis. This perpendicular, the x-axis and the terminal side form a right triangle.</li><li>We've been given that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%29+=+-sqrt%285%29%2F2 ALIGN=MIDDLE  ALT=\"tan%28beta%29+=+-sqrt%285%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since tan is opposite/adjacent, we want to label the opposite site with the numerator of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%285%29%2F2 ALIGN=MIDDLE  ALT=\"-sqrt%285%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the adjacent side with the denominator. But where does the \"=\" go? Answer: Since we are in the 2nd quadrant, the minus should go with the adjacent side (x-axis). So label the opposite side, the perpendicular, as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and label the adjacent side (x-axis) as -2.</li></ul>For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we will use the tan((1/2)x) identity:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28%281%2F2%29x%29+=+sin%28x%29%2F%281%2Bcos%28x%29%29 ALIGN=MIDDLE  ALT=\"tan%28%281%2F2%29x%29+=+sin%28x%29%2F%281%2Bcos%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From this we can see that we will need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28beta%29 ALIGN=MIDDLE  ALT=\"sin%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28beta%29 ALIGN=MIDDLE  ALT=\"cos%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For both sin and cos we need the hypotenuse. So we use the Pythagorean Theorem to find the hypotenuse:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%5E2%2B%28sqrt%285%29%29%5E2+=+h%5E2 ALIGN=MIDDLE  ALT=\"%28-2%29%5E2%2B%28sqrt%285%29%29%5E2+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (where \"h\" represents the hypotenuse)<BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B5+=+h%5E2 ALIGN=MIDDLE  ALT=\"4%2B5+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9+=+h%5E2 ALIGN=MIDDLE  ALT=\"9+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+h ALIGN=MIDDLE  ALT=\"3+=+h\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (ignoring the negative square root of 9 since hypotenuse's are never negative).<br><BR>\n" );
document.write( "Now that we have the hypotenuse, we can find the sin, opposite/hypotenuse, and the cos, adjacent/hypotenuse, of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28beta%29+=+sqrt%285%29%2F3 ALIGN=MIDDLE  ALT=\"sin%28beta%29+=+sqrt%285%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28beta%29+=+%28-2%29%2F3 ALIGN=MIDDLE  ALT=\"cos%28beta%29+=+%28-2%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And finally we can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+sin%28beta%29%2F%281%2Bcos%28beta%29%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+sin%28beta%29%2F%281%2Bcos%28beta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting in the values we found for sin and cos:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%281%2B%28%28-2%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%281%2B%28%28-2%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%283%2F3%2B%28%28-2%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%283%2F3%2B%28%28-2%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%281%2F3%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+%28sqrt%285%29%2F3%29%2F%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+%28%28sqrt%285%29%2F3%29%2F%281%2F3%29%29%283%2F3%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+%28%28sqrt%285%29%2F3%29%2F%281%2F3%29%29%283%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+sqrt%285%29%2F1 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+sqrt%285%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28beta%2F2%29+=+sqrt%285%29 ALIGN=MIDDLE  ALT=\"tan%28beta%2F2%29+=+sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );