document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151811>Question 1151811</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A= 25 degrees<BR>\n" );
document.write( "a=5<BR>\n" );
document.write( "b=11\r<BR>\n" );
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document.write( "Solve the triangle ABC\r<BR>\n" );
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document.write( "Thank you!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773670 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=773670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Angles<BR>\n" );
document.write( "A = 25<BR>\n" );
document.write( "B = unknown<BR>\n" );
document.write( "C = unknown\r<BR>\n" );
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document.write( "Sides<BR>\n" );
document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = unknown\r<BR>\n" );
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document.write( "We have two known sides an angle that is not between the two sides. This is the SSA case which has 3 possibilities<ul><li>Zero triangles are possible</li><li>Exactly one triangle is possible</li><li>Exactly two triangles are possible</li></ul>The SSA case leads to ambiguity.\r<BR>\n" );
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document.write( "Use the law of sines to solve for angle B<BR>\n" );
document.write( "sin(A)/a = sin(B)/b<BR>\n" );
document.write( "sin(25)/5 = sin(B)/11<BR>\n" );
document.write( "0.0845236523481399 = sin(B)/11<BR>\n" );
document.write( "11*0.0845236523481399 = 11*sin(B)/11<BR>\n" );
document.write( "0.929760175829539 = sin(B)<BR>\n" );
document.write( "sin(B) = 0.929760175829539<BR>\n" );
document.write( "arcsin(sin(B)) = arcsin(0.929760175829539)<BR>\n" );
document.write( "B = arcsin(0.929760175829539)  or  B = 180 - arcsin(0.929760175829539)<BR>\n" );
document.write( "B = 68.3974616057519  or  B = 111.602538394248<BR>\n" );
document.write( "B = 68.397  or  B = 111.603\r<BR>\n" );
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document.write( "If B = 68.397, then C = 180-A-B = 180-25-68.397 = 86.603<BR>\n" );
document.write( "If B = 111.603, then C = 180-A-B = 180-25-111.603 = 43.397<BR>\n" );
document.write( "In either case, the value of B produces a positive C value. Therefore, we have two sets of solutions to represent the two possible triangles.\r<BR>\n" );
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document.write( "With this info: C = 86.603, a = 5, b = 11, we can find the unknown side c using the law of cosines<BR>\n" );
document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)<BR>\n" );
document.write( "c^2 = (5)^2 + (11)^2 - 2*(5)*(11)*cos(86.603)<BR>\n" );
document.write( "c^2 = 25 + 121 - 110*0.0592541057795245<BR>\n" );
document.write( "c^2 = 25 + 121 - 6.51795163574769<BR>\n" );
document.write( "c^2 = 139.482048364252<BR>\n" );
document.write( "c = sqrt(139.482048364252)<BR>\n" );
document.write( "c = 11.8102518332274<BR>\n" );
document.write( "c = 11.810\r<BR>\n" );
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document.write( "So one solution consists of<BR>\n" );
document.write( "Angles<BR>\n" );
document.write( "A = 25<BR>\n" );
document.write( "B = 68.397<BR>\n" );
document.write( "C = 86.603<BR>\n" );
document.write( "Sides<BR>\n" );
document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = 11.810\r<BR>\n" );
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document.write( "Use the law of cosines again, but now with C = 43.397<BR>\n" );
document.write( "The values of 'a' and b stay the same<BR>\n" );
document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)<BR>\n" );
document.write( "c^2 = (5)^2 + (11)^2 - 2*(5)*(11)*cos(43.397)<BR>\n" );
document.write( "c^2 = 25 + 121 - 110*0.726610645792925<BR>\n" );
document.write( "c^2 = 25 + 121 - 79.9271710372218<BR>\n" );
document.write( "c^2 = 66.0728289627782<BR>\n" );
document.write( "c = sqrt(66.0728289627782)<BR>\n" );
document.write( "c = 8.12851948160169<BR>\n" );
document.write( "c = 8.129\r<BR>\n" );
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document.write( "The other solution consists of<BR>\n" );
document.write( "Angles<BR>\n" );
document.write( "A = 25<BR>\n" );
document.write( "B = 111.603<BR>\n" );
document.write( "C = 43.397<BR>\n" );
document.write( "Sides<BR>\n" );
document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = 8.129\r<BR>\n" );
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document.write( "In summary, the two possible triangles are<BR>\n" );
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document.write( "<tr><TD  colspan = 2 align = center>Triangle #1</TD><TD  colspan = 2 align = center>Triangle #2</TD></tr>\r\n" );
document.write( "<tr><TD >Angles<BR>\n" );
document.write( "A = 25<BR>\n" );
document.write( "B = 68.397<BR>\n" );
document.write( "C = 86.603</TD><TD >Sides<BR>\n" );
document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = 11.810</TD><TD >Angles<BR>\n" );
document.write( "A = 25<BR>\n" );
document.write( "B = 111.603<BR>\n" );
document.write( "C = 43.397</TD><TD >Sides<BR>\n" );
document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = 8.129</TD></tr>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
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document.write( "Diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width=\"40%\" src = \"https://i.imgur.com/Jh9vjSJ.png\"><BR>\n" );
document.write( "The decimal values are approximate to three decimal places.<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );