Law of Sines Ambiguity

The Math:

The Explanation:

In trigonometry, the Law of Sines, along with the Law of Cosines, are useful tools for calculating unknown sides or angles in a triangle.

In particular, the Law of Sines allows you to find an unknown side or angle given two sides and an angle, or two angles and a side. This is defined by the following relationship:

But the law of sines has an ambigious case where the answer could be one of two angles, either an obtuse angle, or an acute angle. In the ambigious case diagram above, you can think of the 5-lengthed side as an arm that can "swing" to either side, anchored where it meets the 7-lengthed side. Depending on the direction of the 5-lengthed side, angle theta can either be acute (arm swung outward), or obtuse (arm swung inward).

The important thing to remember is that, unless some additional information is given about angle theta, both answers are correct. But, if a diagram is given, or you are otherwise told theta is obtuse or acute, you must calculate the correct angle. Luckily, it's easy to calculate the alternate form of the angle - if the law of sines gives you an acute angle, simply subtract from 180 degrees (pi radians) to obtain the obtuse. This works because of the the periodic, symmetrical nature of the function y=sin(x). As you can see below, both angles result in the same sine value, which is what causes the ambiguity.