This is a free lesson from our course in Trigonometry In this lesson you'll learn with the help of several examples, practice questions with solution concepts of the ambiguous case and approach to problem solving. It will be explained with the help of video presentation and explanation by instructor in own handwriting. If two sides and an angle opposite to one of them are known, it is called the ambiguous case. In such a case we may determine more than one triangle or perhaps no triangles at all. The possible solutions depend on whether the given angle is acute or obtuse. When the Angle is Acute: Let a, b, and angle B be known, and let B be acute. Using the Law of Sines,      sin(A) =  a sin(B)/b. In this five different cases exist. (More text below video...)
Other useful lessons:
 Law of Sines & Cosines Area of a Triangle Using law of Sines and Cosine Formulas

(Continued from above) When the Angle is Obtuse:
Let a, b, and angle B be known, and let B be obtuse. Using the Law of Sines,
sin(A) = a sin(B)/b.
In this three different cases exist.
In case of this being right angle, then you can simply use right triangle solving techniques
Consider the case where two sides of a triangle, say, a and b and an angle B opposite to the side are given, the height of the triangle h can be expressed as b sin B i.e.,
h = b sin B.

• If a < h, then no triangles can be formed.
• If a = h, then only one triangle can be formed.
• If h < a < b, then two triangles can be formed as the side of length a can swing around and form two different triangles.
• If a >= b, then only one triangle can be formed.

For example, in triangle ABC if m B = 64 , side a = 8 cm and b = 12 cm, then plugging the values in h = b sin B will give h = 10.8. In this case since a (= 8) < h (= 10.8). Therefore, no triangles can be formed.

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