document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=312421>Question 312421</A>: <I><SPAN STYLE=\"word-break: break-all;\"> in a triangle, the sides measure 3,5, and 7 what is the measure, in degrees of the largest angle?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #223323 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=223323\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the triangle be ABC with side a opposite angle A, side b opposite angle B, side c opposite angle C.\r<BR>\n" );
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document.write( "Let:\r<BR>\n" );
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document.write( "side a = 3<BR>\n" );
document.write( "side b = 5<BR>\n" );
document.write( "side c = 7\r<BR>\n" );
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document.write( "Since you know all 3 sides, then you can use the law of cosines to find one of the angles.\r<BR>\n" );
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document.write( "Use the formula:\r<BR>\n" );
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document.write( "c^2 = a^2 + b^2 - 2*a*b*cos(C)\r<BR>\n" );
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document.write( "This becomes:\r<BR>\n" );
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document.write( "7^2 = 3^2 + 5^2 - 2*3*5*cos(C)\r<BR>\n" );
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document.write( "Simplify to get:\r<BR>\n" );
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document.write( "49 = 34 - 30*cos(C)\r<BR>\n" );
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document.write( "Subtract 34 from both sides of this equation to get:\r<BR>\n" );
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document.write( "15 = -30*cos(C)\r<BR>\n" );
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document.write( "Solve for cos(C) to get:\r<BR>\n" );
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document.write( "cos(C) = -.5\r<BR>\n" );
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document.write( "Solve for C to get:\r<BR>\n" );
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document.write( "C = 120 degrees.\r<BR>\n" );
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document.write( "You can use the law of cosines again to get another of the angles or you can use the law of sines.\r<BR>\n" );
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document.write( "If you use the law of cosines, then you would solve for:\r<BR>\n" );
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document.write( "b^2 = a^2 + c^2 - 2*a*c*cos(B)\r<BR>\n" );
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document.write( "That would get you:\r<BR>\n" );
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document.write( "25 = 9 + 49 - 42*cos(B) which would get you:\r<BR>\n" );
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document.write( "-42*cos(B) = -33 which would get you:\r<BR>\n" );
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document.write( "cos(B) = .785714286 which would get you:\r<BR>\n" );
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document.write( "B = 38.2132107\r<BR>\n" );
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document.write( "If you used the law of sines to find angle B, you would use the formula:\r<BR>\n" );
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document.write( "b/sin(B) = c/sin(C) which would become:\r<BR>\n" );
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document.write( "5/sin(B) = 7/sin(120) which would become:\r<BR>\n" );
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document.write( "5/sin(B) = 7/.866025404 which would become:\r<BR>\n" );
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document.write( "5/sin(B) = 8.082903769\r<BR>\n" );
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document.write( "Multiply both sides by sin(B) and divide both sides by 8.082903769 to get:\r<BR>\n" );
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document.write( "sin(B) = 5/8.082903769 which would become:\r<BR>\n" );
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document.write( "sin(B) = .618589574 which would become:\r<BR>\n" );
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document.write( "B = 38.2132107 degrees.\r<BR>\n" );
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document.write( "Since we got the second angle using the law of sines and using the law of cosines, then we're probably on the right track.\r<BR>\n" );
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document.write( "Since the sum of the angles of a triangle is equal to 180 degrees, this makes angle A equal to 21.7867893 degrees.\r<BR>\n" );
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document.write( "The three angles of your triangle are:\r<BR>\n" );
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document.write( "A = 21.7867893 degrees<BR>\n" );
document.write( "B = 38.2132107 degrees<BR>\n" );
document.write( "C = 120 degrees\r<BR>\n" );
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document.write( "The largest angle is angle C.\r<BR>\n" );
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document.write( "You can find the third angle in 3 ways.\r<BR>\n" );
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document.write( "1.  Sum of angles of a triangle = 180 degrees.<BR>\n" );
document.write( "2.  Law of Sines<BR>\n" );
document.write( "3.  Law of Cosines\r<BR>\n" );
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document.write( "If you did it correctly, all 3 ways will get you the same answer.\r<BR>\n" );
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document.write( "Using the law of sines, the ratio for all sides divided by the sin of their angles has to be the same.\r<BR>\n" );
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document.write( "Take any one of the sides divided by the sine of their angle to get the ratio.\r<BR>\n" );
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document.write( "Using c/sin(c), we get 7/sin(120) which becomes 8.082903769\r<BR>\n" );
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document.write( "Since side / sine of the angle must equal the ratio, then sine of the angle must equal side / ratio.\r<BR>\n" );
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document.write( "3/8.082903769 gets you .3471153744 which get you an angle of 21.789...\r<BR>\n" );
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document.write( "5/8.082903769  gets you .618589574 which gets you an angle of 38.21...\r<BR>\n" );
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document.write( "All 3 angles follow the law of sines.\r<BR>\n" );
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document.write( "The third angle can also be found using the law of cosines by using the formula:\r<BR>\n" );
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document.write( "a^2 = b^2 + c^2 - 2*b*c*cos(A)\r<BR>\n" );
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document.write( "This becomes:\r<BR>\n" );
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document.write( "3^2 = 5^2 + 7^2 - 2*5*7*cos(A) which simplifies to:\r<BR>\n" );
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document.write( "9 = 25 + 49 - 70*cos(A) which simplifies further to:\r<BR>\n" );
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document.write( "-70*cos(A) = -65 which simplifies further to:\r<BR>\n" );
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document.write( "cos(A) = -65/-70 = .928571429 which gets you an angle of 21.7867893\r<BR>\n" );
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document.write( "All 3 ways get you the same angles as they should.\r<BR>\n" );
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document.write( "It appears that the largest angle is opposite the largest side, as it should in a triangle.\r<BR>\n" );
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document.write( "Solving for all 3 angles confirms that.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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