Determination of Angle from Three Known Sides

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