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An interesting problem arises when two sides and an angle opposite one of them are known. This is called the ambiguous case. A unique triangle is not always determined. The possible solutions depend on whether the given angle is acute or obtuse. When the angle is acute, five possible solutions exist. When the angle is obtuse, three possible solutions exist.

When the Angle is Acute

Let a, b, and B be known, and let B be acute. Using the Law of Sines, sin(A) = . Five different cases exist.

  1. If the side opposite the given angle, b, is shorter than the other given side, a, and less than a certain length, then > 1, and no solution exists, because there exists no angle whose sine is greater than one. Such a case arises when, for example, a = 4, b = 3, and B = 57o.
  2. If the side opposite the given angle is shorter than the other given side, there exists an exact length at which = 1, and A = 90o. Exactly one solution exists, and a right triangle is determined. This occurs, for example, when a = 3, b = 3, and B = 45o.
  3. If the side opposite the given angle is shorter than the other given side, but longer than in case (2), then < 1, and two triangles are determined, one in which A = xo, and one in which A = 180o - xo.
  4. If the side opposite the given angle is equal in length to the other given side, then A = B, and one isosceles triangle is determined.
  5. If the side opposite the given angle is longer than the other given side, then < 1, and one triangle is determined.
Each of these five case is illustrated below.
Figure %: Two sides of an oblique triangle and an angle opposite one of them are given, and the angle is acute.

When the Angle is Obtuse

Let a, b, and B be known, and let B be obtuse. Using the Law of Sines, sin(A) = . Three different cases exist.

  1. If the side opposite the given angle is less than the other given side (b < a), then arcsin() + B > 180o, so there is no solution, and no triangle is determined.
  2. If the side opposite the given angle is equalto the other given side (b = a), then arcsin() + B = 180o, so there is no solution, and, again, no triangle is determined.
  3. If the side opposite the given angle is greater than the other given side, then exactly one triangle is determined. These cases are illustrated below.

Figure %: Two sides of an oblique triangle and an angle opposite one of them are given, and the angle is obtuse.

Summary of Ambiguous Case

In the chart below, the ambiguous case is summarized. The given angle can be either acute or obtuse (if the angle is right, then you can simply use right triangle solving techniques). The side opposite the given angle is either greater than, equal to, or less than the other given side. The chart shows how many triangles can be determined with each possibility, and the case numbers that we used in this section accompany each possibility.

Figure %: In each box of the chart, the number of triangles possibly determined, with the accompanying case # from the text
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