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The  Ambiguous  Case
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Engineering Aid 3 - Beginning Structural engineering guide book
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The value s means one-half the sum of sides a,  b,  and  c,  or Figure 1-29.-Comparison of an ambiguous case triangle to a  standard  triangle. might  satisfy  this  situation.  Both  triangles  shown are   with   given   angle   A   =   30°00´,   given   side a  =  4.00  ft,  and  given  side  c  =  6.00  ft. The best way to determine whether or not the given data for a triangle involves an ambiguous case is to lay out a figure to scale on the basis of the  data,  as  shown  in  figure  1-29.  Suppose,  for example, that the data describes a triangle with angle  A  =  22°00';  side  opposite,  5.40  ft;  and other  side,  14.00  ft.  Lay  off  a  line,  AB,  14.00  ft long (to scale, of course), as shown in the upper triangle of figure 1-29. Use a protractor to lay off a  line  from  A  at  22°00'.  Set  a  compass  to  the graphical  distance  of  5.40  ft  (length  of  side opposite A) and with B as a center, strike an arc. You observe that this arc intersects the line from A at two places. Therefore, the triangle ACB and the triangle ADB both satisfy the data, and you have  an  ambiguous  case. Suppose now that the data describes a triangle with  angle  A  =  35°00´;  side  opposite,  10.00  ft; and other side, 8.00 ft. Lay off the line AB 8.00 ft  long  as  shown  in  the  lower  triangle  of  figure 1-29,  and  lay  off  a  line  from  A  at  35°00’,  Set  a compass  to  10.00  ft  (length  of  side  opposite  A) and with B as a center, strike an arc. This arc will intersect  the  line  from  A  at  only  one  point. Therefore,  only  one  triangle  satisfies  the  data. Determination of Angle from Three Known Sides There are several formulas for determining the size of an angle in a triangle from three known sides. The most convenient involves the versed sine of the angle, which means (1 -cos) of the angle. The  formula  goes  as  follows: For the triangle shown in figure 1-30, you would determine  the  size  of  angle  A  as  follows: The angle with cosine 0.60333 measures (to the nearest minute) 52°53´. Figure 1-30.—Oblique triangle with three sides given and solved by versed sine formula. 1-24







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