Law of Sines - The Ambiguous Case

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

Read more about this topic:  Law Of Sines

Famous quotes containing the words ambiguous and/or case:

    The whole of natural theology ... resolves itself into one simple, though somewhat ambiguous proposition, That the cause or causes of order in the universe probably bear some remote analogy to human intelligence.
    David Hume (1711–1776)

    The real exertion in the case of an opera singer lies not so much in her singing as in her acting of a role, for nearly every modern opera makes great dramatic and physical demands.
    Maria Jeritza (1887–1982)