document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001510>Question 1001510</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the side of a triangle measure 16 centimeters, 13 centimeters, and 11 centimeters, what are the measures of the angles of the triangle<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618769 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=618769\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In a right triangle the longest side is the hypotenuse,<BR>\n" );
document.write( "and if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of the hypotenuse, with <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=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > being the other sides lengths,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the Pythagorean relationship).<BR>\n" );
document.write( "In any triangle ABC,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2-2bc%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the law of cosines).<BR>\n" );
document.write( "(We can also write it as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29 ALIGN=MIDDLE  ALT=\"c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=16 ALIGN=MIDDLE  ALT=\"a=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of side opposite angle A<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%5E2=13%5E3%2B11%5E2-2%2A13%2A11%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"16%5E2=13%5E3%2B11%5E2-2%2A13%2A11%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=256=169%2B121-286%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"256=169%2B121-286%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=286%2Acos%28A%29=169%2B121-256 ALIGN=MIDDLE  ALT=\"286%2Acos%28A%29=169%2B121-256\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=286%2Acos%28A%29=34 ALIGN=MIDDLE  ALT=\"286%2Acos%28A%29=34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29=34%2F286 ALIGN=MIDDLE  ALT=\"cos%28A%29=34%2F286\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=about1.4516 ALIGN=MIDDLE  ALT=\"A=about1.4516\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=about83.17%5Eo ALIGN=MIDDLE  ALT=\"A=about83.17%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "We can also write the law of cosines as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29 ALIGN=MIDDLE  ALT=\"c%5E2=a%5E2%2Bb%5E2-2ab%2Acos%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "To find the measure of another angle,<BR>\n" );
document.write( "we can apply the law of cosines again,<BR>\n" );
document.write( "or we can use the law of sines.<BR>\n" );
document.write( "After that, we find the remaining angle by using the fact that the measures of all 3 angles add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "USING LAW OF COSINES AGAIN:<BR>\n" );
document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the angle opposite the side with length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=13 ALIGN=MIDDLE  ALT=\"b=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >cm.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%5E2=16%5E2%2B11%5E2-2%2A16%2A11%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"13%5E2=16%5E2%2B11%5E2-2%2A16%2A11%2Acos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=169=256%2B121-352%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"169=256%2B121-352%2Acos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=352%2Acos%28B%29=256%2B121-169 ALIGN=MIDDLE  ALT=\"352%2Acos%28B%29=256%2B121-169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=352%2Acos%28B%29=208 ALIGN=MIDDLE  ALT=\"352%2Acos%28B%29=208\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29=208%2F352 ALIGN=MIDDLE  ALT=\"cos%28B%29=208%2F352\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=about0.9386 ALIGN=MIDDLE  ALT=\"B=about0.9386\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=about53.78%5Eo ALIGN=MIDDLE  ALT=\"B=about53.78%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "USING LAW OF SINES FOR A CHANGE:<BR>\n" );
document.write( "If you have the measures of one side and the opposite angle of a triangle,<BR>\n" );
document.write( "you can apply the law of sines, which states<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29%2Fb=sin%28A%29%2Fa=sin%28C%29%2Fc ALIGN=MIDDLE  ALT=\"sin%28B%29%2Fb=sin%28A%29%2Fa=sin%28C%29%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=13 ALIGN=MIDDLE  ALT=\"b=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >cm,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29%2F13=sin%28A%29%2F16 ALIGN=MIDDLE  ALT=\"sin%28B%29%2F13=sin%28A%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29=13sin%28A%29%2F16 ALIGN=MIDDLE  ALT=\"sin%28B%29=13sin%28A%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29=about0.9929 ALIGN=MIDDLE  ALT=\"sin%28A%29=about0.9929\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded), so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29=about13%2A0.9929%2F16 ALIGN=MIDDLE  ALT=\"sin%28B%29=about13%2A0.9929%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=about0.9386 ALIGN=MIDDLE  ALT=\"B=about0.9386\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=about53.78%5Eo ALIGN=MIDDLE  ALT=\"B=about53.78%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=about1.4516%2CB=about0.9386%29 ALIGN=MIDDLE  ALT=\"system%28A=about1.4516%2CB=about0.9386%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=about83.17%5Eo%2CB=about53.78%5Eo%29 ALIGN=MIDDLE  ALT=\"system%28A=about83.17%5Eo%2CB=about53.78%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=pi-about1.4516-about0.9386=about0.7514 ALIGN=MIDDLE  ALT=\"C=pi-about1.4516-about0.9386=about0.7514\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=180%5Eo-83.17%5Eo-about53.78%5Eo=about43.05%5Eo ALIGN=MIDDLE  ALT=\"C=180%5Eo-83.17%5Eo-about53.78%5Eo=about43.05%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );