Then solve for side c using the sine law.
a. The sine of an acute angle is the same as the sine of the supplementary obtuse angle.
Therefore, in solving a triangle under case II by the sine law, two values for the angle are possible and
either value can be taken unless excluded by other conditions in the triangle.
b. In any triangle, only one of the angles can be obtuse. When the given angle is obtuse, both of
the other angles are acute. In Figure 4-20, angle A of the triangle is acute, and side c times the sine of A
equals the line BP. This is the side opposite the given angle in the right triangle ABP. It is also the
altitude of the triangle in question; therefore, BC or BC1 cannot be shorter than the altitude. When side
a is less than side c times the sine of A, the triangle is impossible. When angle A is obtuse, side a must
be longer than side c or the triangle is impossible. When the angle A is acute and the length of the
triangle is shorter than side c but longer than side c times the sine of A (BP), the triangle is ambiguous
and two solutions are possible.
4-12. Case III - Two Sides and the Included Angle. When solving for the third side of a triangle and
two sides and the included angle are known, it is best to use the cosine law.
Figure 4-20. The ambiguous triangle
EN0591
4-28
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