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Solution Triangles

There is one more case, and it is peculiar. This is SSA, where we know two sides and the angle opposite one of them, not the angle between them. Since we know a side and its opposite angle, we can use the Law of Sines to determine the other angle. The problem here is that we are considering angles to go from 00 to 180°, and for such angles the sine is positive and repeats itself. That is, sin(180° - θ) = sin θ. Therefore, unless the angle happens to be 90°, the Law of Sines gives us two possible values, which are supplementary. For that reason, this case is called the ambiguous case. However, there is really nothing wrong here, just that SSA may not specify a triangle uniquely, as a sketch easily reveals. If you know which triangle to choose, and that is part of the statement of the problem, then the problem is as easily solved as any other. We know that a triangle has six parts (or elements). The process of finding the unknown parts of a triangle is called the solution of triangle. If three parts of a triangle (at least one of which is side) are given, then the other parts can be found.

In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information. In particular:

• If one knows two of the angles one can find the third by using the fact that the sum of the three must be 180°.
• (SSS) If one knows the lengths of the three sides, one can find the three angles by using the law of cosines.
• (SAS) If one knows the lengths of two of the sides and the angle between them, one can find the length of the third side by using the law of cosines.
• (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the third length and the other angles by using the law of sines, in some cases up to a choice between two possible solutions.
• (SAA) If one knows the length of one side and at least two of the angles, one can find the lengths of the other sides by using the law of sines.
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