document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189698>Question 1189698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hey, Can anyone help me with this question I am really confused. Thank you.\r<BR>\n" );
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document.write( "A surveyor is mapping a triangular field. From one corner he sights between the two opposite corners and finds an angle of 56°. He knows that he length of the opposite side of the field is 60m while the side to his left is 70m. The surveyor creates one drawing of the field and sends the single drawing and a bill to the client. Should the client pay the bill? Explain your reasoning </SPAN>\n" );
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document.write( "Let's say the surveyor is at point A. <BR>\n" );
document.write( "He measures an angle of 56 degrees at this location when looking at the other corners B and C. <BR>\n" );
document.write( "So angle A = 56.\r<BR>\n" );
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document.write( "The side opposite angle A is side 'a'<BR>\n" );
document.write( "b is opposite angle B<BR>\n" );
document.write( "c is opposite angle C\r<BR>\n" );
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document.write( "The surveyor figured out that a = 60 meters and b = 70 meters. Assume side b is to the left of point A.\r<BR>\n" );
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document.write( "We have these properties<BR>\n" );
document.write( "angle A = 56 degrees<BR>\n" );
document.write( "side a = 60 meters<BR>\n" );
document.write( "side b = 70 meters\r<BR>\n" );
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document.write( "Let's use the law of sines to see if we can figure out angle B<BR>\n" );
document.write( "sin(A)/a = sin(B)/b<BR>\n" );
document.write( "sin(56)/60 = sin(B)/70<BR>\n" );
document.write( "sin(B) = 70*sin(56)/60<BR>\n" );
document.write( "sin(B) = 0.9672105 approximately<BR>\n" );
document.write( "B = arcsin(0.9672105) or B = 180-arcsin(0.9672105)<BR>\n" );
document.write( "B = 75.287079 or B = 180-75.287079<BR>\n" );
document.write( "B = 75.287079 or B = 104.712921\r<BR>\n" );
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document.write( "If B = 75.287079, then,<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-56-75.287079<BR>\n" );
document.write( "C = 48.712921<BR>\n" );
document.write( "which forms a valid triangle.\r<BR>\n" );
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document.write( "If B = 104.712921, then<BR>\n" );
document.write( "C = 180-A-B<BR>\n" );
document.write( "C = 180-56-104.712921<BR>\n" );
document.write( "C = 19.287079<BR>\n" );
document.write( "which forms a different triangle\r<BR>\n" );
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document.write( "So the info that the surveyor currently has is <u>not</u> enough information. <BR>\n" );
document.write( "Two triangles are possible, which means the surveyor needs to go back and do at least one more measurement.\r<BR>\n" );
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document.write( "The client should <u>not</u> pay the surveyor (at least not until the surveyor completes the task). \r<BR>\n" );
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document.write( "For more information, search out \"SSA ambiguous case\" or \"law of sines ambiguous case\".<BR>\n" );
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