document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172758>Question 1172758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The point P on the side BC of triangle ABC divides BC in the ratio 1:2 i.e. BP:PC=1:2 , angle ABC =45° , angle APC=60° , <BR>\n" );
document.write( "calculate angle ACB. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #797928 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=797928\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There is probably a relatively simple solution method, because the answer turns out to be \"nice\".  But I'm not seeing an easy solution.  So....<br><BR>\n" );
document.write( "We can arbitrarily let BP=1 and PC=2.<br><BR>\n" );
document.write( "With angle ABC 45 degrees and angle APC 60 degrees, angle APB is 120 degrees; that makes angle BAP 15 degrees.<br><BR>\n" );
document.write( "Let x be the measure of angle ACB that we are looking for; then the measure of angle CAP is 120-x.<br><BR>\n" );
document.write( "Let y be the length of AP.<br><BR>\n" );
document.write( "Then in triangle BAP the law of sines gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fsin%2815%29+=+y%2Fsin%2845%29 ALIGN=MIDDLE  ALT=\"1%2Fsin%2815%29+=+y%2Fsin%2845%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sin%2845%29%2Fsin%2815%29 ALIGN=MIDDLE  ALT=\"y+=+sin%2845%29%2Fsin%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And in triangle APC the law of sines gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Fsin%28x%29+=+2%2Fsin%28120-x%29 ALIGN=MIDDLE  ALT=\"y%2Fsin%28x%29+=+2%2Fsin%28120-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2sin%28x%29%2Fsin%28120-x%29 ALIGN=MIDDLE  ALT=\"y+=+2sin%28x%29%2Fsin%28120-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we have two expressions for y -- one a constant and the other an expression in x.  Set them equal to each other:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sin%28x%29%2Fsin%28120-x%29+=+sin%2845%29%2Fsin%2815%29 ALIGN=MIDDLE  ALT=\"2sin%28x%29%2Fsin%28120-x%29+=+sin%2845%29%2Fsin%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Graphing the two expressions on a graphing calculator shows that x, the measure of angle ACB we are looking for, is 75 degrees.<br><BR>\n" );
document.write( "ANSWER: angle ACB is 75 degrees.<br><BR>\n" );
document.write( "I will be watching with curiosity to see if another tutor comes up with a simpler solution method....<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );