The Ambiguous Case
When using the law of sines to solve triangles, there exists an ambiguous case where two separate triangles can be constructed (i.e., there are two different possible solutions to the triangle).
Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:
- The only information known about the triangle is the angle A and the sides a and b
- The angle A is acute (i.e., A < 90°).
- The side a is shorter than the side b (i.e., a < b).
- The side a is longer than the altitude of a right angled triangle with angle A and hypotenuse b (i.e., a > b sin A).
Given all of the above premises are true, the angle B may be acute or obtuse; meaning, one of the following is true:
or
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