document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1065976>Question 1065976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given that theta is an acute angle and sin(theta) = 17/37, find the exact value of sec(theta)\r<BR>\n" );
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document.write( "Use the exact values for the sine and cosine of both pi/4 and 2pi/3, and the angle sum identity for cosine, fine the exact value of cos(11pi/12)\r<BR>\n" );
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document.write( "use the exact value of cos(11pi/6) and the half-angle identity for sine to find the exact value of sin(11pi/12)\r<BR>\n" );
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document.write( "I believe to do this question I need to find which quadrants the angles lie in, but beyond that I am just not sure what to do! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #681162 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=681162\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> </BLOCKQUOTE><FONT COLOR=red>Given that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an acute angle and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+17%2F37 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+17%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" >, find the exact value of sec(theta) </FONT><BR><BR>\n" );
document.write( "THE SEVENTH GRADER'S POINT OF VIEW:<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an acute angle, it could be an angle in a right triangle, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29=17%2F37 ALIGN=MIDDLE  ALT=\"sin%28theta%29=17%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be seen as a trigonometric ratio,<BR>\n" );
document.write( "relating the lengths of the hypotenuse and opposite side in a right triangle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C200%2C-1%2C35%2C-2%2C18%2C%0D%0Agreen%28rectangle%2832.86%2C0%2C31.86%2C1%29%29%2C%0D%0Atriangle%280%2C0%2C32.86%2C0%2C32.86%2C17%29%2C%0D%0Ared%28arc%280%2C0%2C10%2C10%2C-27.35%2C0%29%29%2C%0D%0Alocate%2816%2C0%2Cx%29%2Clocate%2833%2C9%2C17%29%2C%0D%0Alocate%2816.43%2C8.5%2C37%29%2Clocate%283%2C1.5%2Cred%28theta%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . With that point of view you do not need to know about quadrants,<BR>\n" );
document.write( "or about trigonometric functions.<BR>\n" );
document.write( "Then you would find the length of the missing side using the Pythagorean theorem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B17%5E2=37%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B17%5E2=37%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B289=1369 ALIGN=MIDDLE  ALT=\"x%5E2%2B289=1369\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=1369-289 ALIGN=MIDDLE  ALT=\"x%5E2=1369-289\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=1080 ALIGN=MIDDLE  ALT=\"x%5E2=1080\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%281080%29=sqrt%2836%2A30%29=sqrt%2836%29%2Asqrt%2830%29=6sqrt%2830%29 ALIGN=MIDDLE  ALT=\"x=sqrt%281080%29=sqrt%2836%2A30%29=sqrt%2836%29%2Asqrt%2830%29=6sqrt%2830%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then you would find <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;\" > as the ratio (adjacent side)/hypotenuse:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=highlight%286sqrt%2830%29%2F37=about0.8882%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29=highlight%286sqrt%2830%29%2F37=about0.8882%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE HIGH SCHOOLER'S POINT OF VIEW:<BR>\n" );
document.write( "When we need to extend the definitions of trigonometric ratios to larger angles, measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or more, which cannot be part of a right triangle,<BR>\n" );
document.write( "we start talking about quadrants and trigonometric functions.<BR>\n" );
document.write( "At that point we say that<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an acute angle, it si in the first quadrant,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ctheta%3C90%5Eo ALIGN=MIDDLE  ALT=\"0%3Ctheta%3C90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ctheta%3Cpi%2F2 ALIGN=MIDDLE  ALT=\"0%3Ctheta%3Cpi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and all trigonometric functions are positive.<BR>\n" );
document.write( "Then, we use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+17%2F37 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+17%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to find <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;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2817%2F37%29%5E2%2Bcos%5E2%28theta%29=1 ALIGN=MIDDLE  ALT=\"%2817%2F37%29%5E2%2Bcos%5E2%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%5E2%2F37%5E2%2Bcos%5E2%28theta%29=1 ALIGN=MIDDLE  ALT=\"17%5E2%2F37%5E2%2Bcos%5E2%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29=1-17%5E2%2F37%5E2 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29=1-17%5E2%2F37%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29=%2837%5E2-17%5E2%29%2F37%5E2 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29=%2837%5E2-17%5E2%29%2F37%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29=%281369-289%29%2F37%5E2 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29=%281369-289%29%2F37%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29=1080%2F37%5E2 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29=1080%2F37%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29=sqrt%281080%2F37%5E2%29=sqrt%281080%29%2F37=highlight%286sqrt%2830%29%2F37=about0.8882%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE SECOND QUESTION:<BR>\n" );
document.write( "That question is beyond the 7th grade math.<BR>\n" );
document.write( "At this point in his/her studies,<BR>\n" );
document.write( "a student should be resigned to the fact that radians are here to stay,<BR>\n" );
document.write( "because degrees are for people whose career options will be limited.<BR>\n" );
document.write( "If you are there, you know that a right angle measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2%7D%7D%2C%0D%0Aand+you+know+where+each+quadrant+begins+and+ends.%0D%0AThere+are+some+trigonometric+definitions+and+identities+that%0D%0Aeveryone+dealing+with+trigonometry+should+remember%3A%0D%0A%7B%7B%7Btan%28theta%29=sin%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=cot%28theta%29=1%2Ftan%28theta%29 ALIGN=MIDDLE  ALT=\"cot%28theta%29=1%2Ftan%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=sec%28theta%29=1%2Fcos%28theta%29 ALIGN=MIDDLE  ALT=\"sec%28theta%29=1%2Fcos%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=csc%28theta%29=1%2Fsin%28theta%29 ALIGN=MIDDLE  ALT=\"csc%28theta%29=1%2Fsin%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=sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For the rest, you should be allowed to look up the formulas.<BR>\n" );
document.write( "If your life is such that you need to look up those formulas often,<BR>\n" );
document.write( "you will eventually remember them.<BR>\n" );
document.write( "I just search online for \"trig identities\" and trust Wikipedia.<BR>\n" );
document.write( "It says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha+%2B+beta%29=sin%28alpha%29%2Acos%28beta%29+%2B+sin%28beta%29%2Acos%28alpha%29 ALIGN=MIDDLE  ALT=\"sin%28alpha+%2B+beta%29=sin%28alpha%29%2Acos%28beta%29+%2B+sin%28beta%29%2Acos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha+%2B+beta%29=cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%29 ALIGN=MIDDLE  ALT=\"cos%28alpha+%2B+beta%29=cos%28alpha%29%2Acos%28beta%29+-+sin%28alpha%29%2Asin%28beta%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%2F2%29=%281-cos%28theta%29%29%2F2 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%2F2%29=%281-cos%28theta%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%28theta%29%2F2=%281%2Bcos%28theta%29%29%2F2 ALIGN=MIDDLE  ALT=\"cos%5E2%28theta%29%2F2=%281%2Bcos%28theta%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "That is all I need.<BR>\n" );
document.write( "I remember the exact values of the trigonometric functions for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because those are the angles in<BR>\n" );
document.write( "the half of a square cut along the diagonal,<BR>\n" );
document.write( "and an equilateral triangle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C1.1%2C-0.1%2C1.1%2C%0D%0Arectangle%280%2C0%2C1%2C1%29%2Ctriangle%280%2C0%2C1%2C0%2C1%2C1%29%2C%0D%0Arectangle%281%2C0%2C0.95%2C0.05%29%2Cred%28arc%280%2C0%2C0.4%2C0.4%2C-45%2C0%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.35%2C0.35%2C-90%2C-45%29%29%2Cred%28arc%281%2C1%2C0.35%2C0.35%2C90%2C135%29%29%2C%0D%0Ared%28arc%281%2C1%2C0.38%2C0.38%2C135%2C180%29%29%2Clocate%280.2%2C0.15%2Cred%28pi%2F4%29%29%2C%0D%0Alocate%280.45%2C0%2C1%29%2Clocate%281.03%2C0.55%2C1%29%2C%0D%0Alocate%280.5%2C0.5%2Csqrt%282%29%29%2Crectangle%280%2C1%2C0.05%2C0.95%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F4%29=cos%28pi%2F4%29=1%2Fsqrt%282%29=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28pi%2F4%29=cos%28pi%2F4%29=1%2Fsqrt%282%29=sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28sin%28pi%2F3%29=sqrt%281%5E2-%281%2F2%29%5E2%29=sqrt%281-1%2F4%29=sqrt%283%2F4%29=sqrt%283%29%2F2%2Ccos%28pi%2F3%29=1%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <--- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.6%2C0.6%2C-0.2%2C1%2C%0D%0Atriangle%28-0.5%2C0%2C0.5%2C0%2C0%2C0.866%29%2Ctriangle%280%2C0%2C0.5%2C0%2C0%2C0.866%29%2C%0D%0Arectangle%280%2C0%2C-0.05%2C0.05%29%2Cred%28arc%28-0.5%2C0%2C0.3%2C0.3%2C-60%2C0%29%29%2C%0D%0Ared%28arc%280.5%2C0%2C0.3%2C0.3%2C-180%2C-120%29%29%2Cred%28arc%280%2C0.866%2C0.3%2C0.3%2C60%2C120%29%29%2C%0D%0Alocate%28-0.27%2C0%2C1%2F2%29%2Clocate%280.23%2C0%2C1%2F2%29%2Clocate%28-0.25%2C0.433%2C1%29%2C%0D%0Alocate%280.23%2C0.433%2C1%29%2C+locate%280.03%2C0.4%2Csqrt%283%29%2F2%29%2C%0D%0Alocate%28-0.35%2C0.15%2Cred%28pi%2F3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F3=pi-pi%2F3%29 ALIGN=MIDDLE  ALT=\"2pi%2F3=pi-pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28450%2C255%2C-1.2%2C1.2%2C-0.16%2C1.2%2C%0D%0Agrid%280%29%2Cred%28circle%280%2C0%2C1%29%29%2Ctriangle%280%2C0%2C0.5%2C0%2C0.5%2C0.866%29%2C%0D%0Atriangle%280%2C0%2C-0.5%2C0%2C-0.5%2C0.866%29%2Clocate%280.52%2C0.5%2Csin%28pi%2F3%29%29%2C%0D%0Alocate%28-0.48%2C0.5%2Csin%282pi%2F3%29%29%2Cred%28arc%280%2C0%2C0.3%2C0.3%2C-60%2C0%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.3%2C0.3%2C-180%2C-120%29%29%2Clocate%280.15%2C0.2%2Cred%28pi%2F3%29%29%2C%0D%0Agreen%28arc%280%2C0%2C0.6%2C0.6%2C-120%2C0%29%29%2Clocate%280.02%2C0.47%2Cgreen%282pi%2F3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28sin%282pi%2F3%29=sin%28pi%2F3%29=sqrt%283%29%2F2%2Ccos%282pi%2F3%29=-cos%28pi%2F3%29=-1%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now, we can </BLOCKQUOTE><FONT COLOR=red>use the exact values for the sine and cosine of both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F3 ALIGN=MIDDLE  ALT=\"2pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and the angle sum identity for cosine, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28to%22 ALIGN=MIDDLE  ALT=\"%22%28to%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22find+%29%22 ALIGN=MIDDLE  ALT=\"%22find+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2811pi%2F12%29 ALIGN=MIDDLE  ALT=\"cos%2811pi%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></FONT><BR> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11pi%2F12=%288%2B3%29pi%2F12=8pi%2F12%2B3pi%2F12=2pi%2F3%2Bpi%2F4 ALIGN=MIDDLE  ALT=\"11pi%2F12=%288%2B3%29pi%2F12=8pi%2F12%2B3pi%2F12=2pi%2F3%2Bpi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so applying the sum identity for cosine<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2811pi%2F12%29=cos%282pi%2F3%29%2Acos%28pi%2F4%29-sin%282pi%2F3%29%2Asin%28pi%2F4%29=%28-1%2F2%29%28sqrt%282%29%2F2%29-%28sqrt%283%29%2F2%29%28sqrt%282%29%2F2%29=%28-sqrt%282%29-sqrt%282%29%2Asqrt%283%29%29%2F4=highlight%28-sqrt%282%29%281%2Bsqrt%283%29%29%2F4=-%28sqrt%282%29%2Bsqrt%286%29%29%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "As for using </BLOCKQUOTE><FONT COLOR=red>the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2811pi%2F6%29 ALIGN=MIDDLE  ALT=\"cos%2811pi%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the half-angle identity for sine to find the exact value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2811pi%2F12%29 ALIGN=MIDDLE  ALT=\"sin%2811pi%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></FONT><BR> ,<BR>\n" );
document.write( "we would have to figure out how to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2811pi%2F6%29 ALIGN=MIDDLE  ALT=\"cos%2811pi%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > first.<BR>\n" );
document.write( "And, why would we do that, instead of using the already found value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2811pi%2F12%29 ALIGN=MIDDLE  ALT=\"cos%2811pi%2F12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the trigonometric identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "Well, It is not that hard.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11pi%2F6=%2812-1%29pi%2F6=12pi%2F6-pi%2F6=2pi-pi%2F6 ALIGN=MIDDLE  ALT=\"11pi%2F6=%2812-1%29pi%2F6=12pi%2F6-pi%2F6=2pi-pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28255%2C450%2C-0.2%2C1.16%2C-1.2%2C1.2%2C%0D%0Agrid%280%29%2Cred%28circle%280%2C0%2C1%29%29%2Ctriangle%280%2C0%2C0.866%2C0.5%2C0.866%2C-0.5%29%2C%0D%0Arectangle%280.866%2C0%2C0.816%2C0.05%29%2Cred%28arc%280%2C0%2C0.6%2C0.6%2C-30%2C0%29%29%2C%0D%0Ared%28arc%280%2C0%2C0.4%2C0.4%2C0%2C30%29%29%2Clocate%280.3%2C0.2%2Cred%28pi%2F6%29%29%2C%0D%0Agreen%28arc%280%2C0%2C0.35%2C0.35%2C30%2C360%29%29%2Clocate%280.05%2C-0.16%2Cgreen%2811pi%2F6%29%29%2C%0D%0Alocate%280.433%2C0.25%2C1%29%2Clocate%280.87%2C0.3%2Csqrt%283%29%2F2%29%2Clocate%280.45%2C0%2C1%2F2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and we see that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28sin%2811pi%2F6%29=-sin%28pi%2F6%29=-1%2F2%2Ccos%2811pi%2F6%29=cos%28pi%2F6%29=sqrt%283%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using that, and the half-angle identity for sine, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%2F2%29=%281-cos%28theta%29%29%2F2 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%2F2%29=%281-cos%28theta%29%29%2F2\"   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%2811pi%2F12%29=sin%5E2%28%2811pi%2F6%29%2F2%29=%281-cos%2811pi%2F6%29%29%2F2=%281-sqrt%283%29%2F2%29%2F2=%282-sqrt%283%29%29%2F4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11pi%2F6 ALIGN=MIDDLE  ALT=\"11pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > was almost a full turn, almost at the far end of the fourth quadrant,<BR>\n" );
document.write( "we know that half of that, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11pi%2F12 ALIGN=MIDDLE  ALT=\"11pi%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, would be almost,<BR>\n" );
document.write( "almost at the end of the second quadrant.<BR>\n" );
document.write( "We also know that in the second quadrant sine is positive, and cosine is negative.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2811pi%2F12%29=sqrt%28%282-sqrt%283%29%29%2F4%29=highlight%28sqrt%282-sqrt%283%29%29%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );