document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1073587>Question 1073587</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The &#64257;rst and second terms of a geometric progression are cos &#952; and sin &#952; respectively.<BR>\n" );
document.write( "(a) Find the common ration r and the third term of this geometric progression in terms of &#952;.<BR>\n" );
document.write( "(b) Given that the &#64257;rst and third terms of this geometric progression and tan &#952; are three consecutive terms of an arithmetic progression, &#64257;nd the general solution for &#952;. Give your answer in radians correct to two decimal places.<BR>\n" );
document.write( "(c) Suppose that 0 < r < 1. Find the sum to in&#64257;nity of the geometric progression in this case. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #688502 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=688502\">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=cos%28theta%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the first term in a geometric progression<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the second term term in the same geometric progression.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) The common ratio, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > in a geometric progression is the ratio of one term to the next,<BR>\n" );
document.write( "one term divided by the term before. In this case<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28r=sin%28theta%29%2Fcos%28theta%29=tan%28theta%29%29 ALIGN=MIDDLE  ALT=\"highlight%28r=sin%28theta%29%2Fcos%28theta%29=tan%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The third term is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29%2Ar=sin%28theta%29%2Atan%28theta%29=sin%28theta%29%2Asin%28theta%29%2Fcos%28theta%29=sin%5E2%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) So the first three terms of the arithmetic progression in this problem are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=cos%28theta%29 ALIGN=MIDDLE  ALT=\"a%5B1%5D=cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=sin%5E2%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"a%5B2%5D=sin%5E2%28theta%29%2Fcos%28theta%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=a%5B3%5D=tan%28theta%29=sin%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"a%5B3%5D=tan%28theta%29=sin%28theta%29%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The common difference is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=a%5B2%5D-a%5B1%5D=sin%5E2%28theta%29%2Fcos%28theta%29-cos%28theta%29 ALIGN=MIDDLE  ALT=\"d=a%5B2%5D-a%5B1%5D=sin%5E2%28theta%29%2Fcos%28theta%29-cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but it is also<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=a%5B3%5D-a%5B2%5D=sin%28theta%29%2Fcos%28theta%29-sin%5E2%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"d=a%5B3%5D-a%5B2%5D=sin%28theta%29%2Fcos%28theta%29-sin%5E2%28theta%29%2Fcos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, our equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Fcos%28theta%29-cos%28theta%29=sin%28theta%29%2Fcos%28theta%29-sin%5E2%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Fcos%28theta%29-cos%28theta%29=sin%28theta%29%2Fcos%28theta%29-sin%5E2%28theta%29%2Fcos%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sin%5E2%28theta%29-cos%5E2%28theta%29%29%2Fcos%28theta%29=%28sin%28theta%29-sin%5E2%28theta%29%29%2Fcos%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29-cos%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29-cos%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2B1-cos%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29%2B1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2B1-cos%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Bsin%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29%2B1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bsin%5E2%28theta%29=sin%28theta%29-sin%5E2%28theta%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Bsin%5E2%28theta%29%2Bsin%5E2%28theta%29=sin%28theta%29%2B1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bsin%5E2%28theta%29%2Bsin%5E2%28theta%29=sin%28theta%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sin%5E2%28theta%29=sin%28theta%29%2B1 ALIGN=MIDDLE  ALT=\"3sin%5E2%28theta%29=sin%28theta%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I am tired of writing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so many times.<BR>\n" );
document.write( "I will abbreviate it as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sin%28theta%29 ALIGN=MIDDLE  ALT=\"x=sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for a while.<BR>\n" );
document.write( "(It's what the teacher would call a change of variable).<BR>\n" );
document.write( "Now the equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2=x%2B1 ALIGN=MIDDLE  ALT=\"3x%5E2=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-x-1=0 ALIGN=MIDDLE  ALT=\"3x%5E2-x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and I can solve it using the quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-1%29+%2B-+sqrt%28%28-1%29%5E2-4%2A3%2A%28-1%29%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-1%29+%2B-+sqrt%28%28-1%29%5E2-4%2A3%2A%28-1%29%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+sqrt%281%2B12%29%29%2F6+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+sqrt%281%2B12%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+sqrt%2813%29%29%2F6+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+sqrt%2813%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That yields two solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sin%28theta%29 ALIGN=MIDDLE  ALT=\"x=sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The approximate values are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2Bsqrt%2813%29%29%2F6=about0.76759 ALIGN=MIDDLE  ALT=\"x+=+%281%2Bsqrt%2813%29%29%2F6=about0.76759\"   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+=+%281-sqrt%2813%29%29%2F6=about-0.43426 ALIGN=MIDDLE  ALT=\"x+=+%281-sqrt%2813%29%29%2F6=about-0.43426\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "since both are between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "they both could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Using the inverse function of sine, in radians, I find that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%280.87508%29=%281%2Bsqrt%2813%29%29%2F6 ALIGN=MIDDLE  ALT=\"sin%280.87508%29=%281%2Bsqrt%2813%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-0.44921%29=%281%2Bsqrt%2813%29%29%2F6 ALIGN=MIDDLE  ALT=\"sin%28-0.44921%29=%281%2Bsqrt%2813%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The inverse sine function gives you the solutions in quadrants 1 and 4,<BR>\n" );
document.write( "but we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi-theta%29=sin%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28pi-theta%29=sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so that would give us two more solutions:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.14159-0.87508=2.26652 ALIGN=MIDDLE  ALT=\"3.14159-0.87508=2.26652\"   DISABLED_style= \"max-width:90%; height:auto;\" > in quadrant 2, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3.14159-%28-0.44921%29=3.14159%2B0.44921=3.59081 ALIGN=MIDDLE  ALT=\"3.14159-%28-0.44921%29=3.14159%2B0.44921=3.59081\"   DISABLED_style= \"max-width:90%; height:auto;\" > in quadrant 3,<BR>\n" );
document.write( "for a total of 4 solutions, one per quadrant.<BR>\n" );
document.write( "There is an infinity number of other solutions,<BR>\n" );
document.write( "because adding an integer number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > of whole turns to an angle,<BR>\n" );
document.write( "you get a co-terminal angle that has the same values for all its trigonometric functions.<BR>\n" );
document.write( "So, the general solutions are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=0.88%2Bk2pi%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=0.88%2Bk2pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=2.27%2Bk2pi%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=2.27%2Bk2pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=-0.451%2Bk2pi%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=-0.451%2Bk2pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28theta=3.59%2Bk2pi%29 ALIGN=MIDDLE  ALT=\"highlight%28theta=3.59%2Bk2pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "There are other ways to write the solution.<BR>\n" );
document.write( "I could use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2B%28-0.44921%29=5.83397 ALIGN=MIDDLE  ALT=\"2pi%2B%28-0.44921%29=5.83397\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the quadrant 4 solution, if I di not like to see negative numbers.<BR>\n" );
document.write( "I could use 3.14 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but then as {{k}}} increases the general solution would not be \"correct to two decimal places.\"<BR>\n" );
document.write( "I could have two formulas if I consolidate quadrants,<BR>\n" );
document.write( "because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi-theta ALIGN=MIDDLE  ALT=\"pi-theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be written<BR>\n" );
document.write( "(in one cumbersome expression) as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2+%2B-+%28theta-pi%2F2%29 ALIGN=MIDDLE  ALT=\"pi%2F2+%2B-+%28theta-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "(c) The sum to infinity of a geometric progression<BR>\n" );
document.write( "with first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2Cr%2C1 ALIGN=MIDDLE  ALT=\"0%2Cr%2C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F%281-r%29 ALIGN=MIDDLE  ALT=\"b%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Is this part supposed to be connected to part (b)?<BR>\n" );
document.write( "I suppose so. This is almost as evil a problem as the high school comprehensive finals I used to have in Uruguay.<BR>\n" );
document.write( "We could conceive that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cr=tan%28theta%29=sin%28theta%29%2Fcos%28theta%29%3C1 ALIGN=MIDDLE  ALT=\"0%3Cr=tan%28theta%29=sin%28theta%29%2Fcos%28theta%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means quadrants 1 or 3 for a positive tangent, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29%3Ccos%28theta%29 ALIGN=MIDDLE  ALT=\"sin%28theta%29%3Ccos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29%3Csqrt%282%29%2F2=about0.707 ALIGN=MIDDLE  ALT=\"sin%28theta%29%3Csqrt%282%29%2F2=about0.707\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Considering that and the solutions to part (b) above,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+%281%2Bsqrt%2813%29%29%2F6=about0.76759 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+%281%2Bsqrt%2813%29%29%2F6=about0.76759\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not work,<BR>\n" );
document.write( "but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29++=+%281-sqrt%2813%29%29%2F6=about-0.43426 ALIGN=MIDDLE  ALT=\"sin%28theta%29++=+%281-sqrt%2813%29%29%2F6=about-0.43426\"   DISABLED_style= \"max-width:90%; height:auto;\" > would work<BR>\n" );
document.write( "for a quadrant 3 angle, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=+-+sqrt%281-sin%5E2%28theta%29%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=+-+sqrt%281-sin%5E2%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Hopefully an approximate solution would be OK.<BR>\n" );
document.write( "In that case, using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=tan%283.59081%29=0.48209 ALIGN=MIDDLE  ALT=\"r=tan%283.59081%29=0.48209\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=cos%28theta%29=cos%283.59081%29=-0.90079 ALIGN=MIDDLE  ALT=\"b=cos%28theta%29=cos%283.59081%29=-0.90079\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum=%28-0.90079%29%2F%281-0.48209%29=about ALIGN=MIDDLE  ALT=\"sum=%28-0.90079%29%2F%281-0.48209%29=about\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-1.739%29 ALIGN=MIDDLE  ALT=\"highlight%28-1.739%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );