The value s means one-half the sum of sidesa, b, and c, orFigure 1-29.-Comparison of an ambiguous case triangle toa standard triangle.might satisfy this situation. Both triangles shownare with given angle A = 30°00´, given sidea = 4.00 ft, and given side c = 6.00 ft.The best way to determine whether or not thegiven data for a triangle involves an ambiguouscase is to lay out a figure to scale on the basis ofthe data, as shown in figure 1-29. Suppose, forexample, that the data describes a triangle withangle A = 22°00'; side opposite, 5.40 ft; andother side, 14.00 ft. Lay off a line, AB, 14.00 ftlong (to scale, of course), as shown in the uppertriangle of figure 1-29. Use a protractor to lay offa line from A at 22°00'. Set a compass to thegraphical distance of 5.40 ft (length of sideopposite A) and with B as a center, strike an arc.You observe that this arc intersects the line fromA at two places. Therefore, the triangle ACB andthe triangle ADB both satisfy the data, and youhave an ambiguous case.Suppose now that the data describes a trianglewith angle A = 35°00´; side opposite, 10.00 ft;and other side, 8.00 ft. Lay off the line AB 8.00ft long as shown in the lower triangle of figure1-29, and lay off a line from A at 35°00’, Set acompass to 10.00 ft (length of side opposite A)and with B as a center, strike an arc. This arc willintersect the line from A at only one point.Therefore, only one triangle satisfies the data.Determination of Anglefrom Three Known SidesThere are several formulas for determining thesize of an angle in a triangle from three knownsides. The most convenient involves the versed sineof the angle, which means (1 -cos) of the angle.The formula goes as follows:For the triangle shown in figure 1-30, you woulddetermine the size of angle A as follows:The angle with cosine 0.60333 measures (to thenearest minute) 52°53´.Figure 1-30.—Oblique triangle with three sides given andsolved by versed sine formula.1-24
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