When using the law of sines, why can the SSA case result in zero, one, or two triangles?
1 Answer
Explanation:
When we have two sides and an angle there are two cases: If the angle is the included angle, formed by the two sides, the triangle is uniquely determined and the path to solving starts with the Law Of Cosines.
The other case is the angle isn't the included angle, succinctly described as SSA. Say we have sides
Now we have the three cases mentioned in the question. First we note none of the quantities are negative; for triangle angles the sine is always positive.
The expression on the right may evaluate to more than 1. In that case, there's no such triangle that satisfies the givens.
Or the expression may evaluate to exactly 1,
Finally, the most common case is
The two possible angles for A give two possible triangles ABC which satisfy the given SSA.
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