# Why are there two possible triangles when given SAS?

I gave my trigonometry students the following example: Solve $\triangle ABC\$ , where AC=0.923, AB=.387, and $\measuredangle A\ = 43.33^\circ\$. First I found BC using the law of cosines, then I found $\measuredangle B\$ using the law of sines, and $\measuredangle C\$ using the sum of the interior angles of the triangle.
Some of my students found BC first, then found $\measuredangle C\$ and got a different value then if they had solved it my way. Why is this? I was under the impression that the ambiguous case existed when you're in the situation of SSA. Note: the textbook says that the convention is to solve for the largest angle first. Why is this?

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I believe the error that you or your students made is in assuming that $\arcsin (\sin(\theta))=\theta$. This is false for an obtuse angle. In this case, since $AB < AC$, you know that $\measuredangle C$ is acute, so it sounds like your students' method is correct.

When using the law of cosines to find an angle, there is no problem, because $\arccos (\cos(\theta))=\theta$ is correct for the angles in a triangle. The law of sines is easier to use however, so it is nice to reduce to the acute case. That is why if you have SSS, it is a good idea to find the largest angle (opposite the largest side) first, because then you can safely use $\arcsin$ to find the other angles.

On the other hand, if you know in advance that the angle $\theta$ you're trying to find is obtuse, you can use $\theta=\pi - \arcsin(\sin(\theta))$. The ambiguity from the $\sin$ function is that supplementary angles have the same $\sin$. Given SAS, you can first find the other side using law of cosines, and next use law of sines to find the angle opposite one of the shorter sides, because you know it will be acute.

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Yes, you're right about the assumption. It makes sense to find the largest angle by using the law of cosines. Thank you. – kuttaka Feb 3 '11 at 20:21

Does the value they found for $\angle C$ actually solve the problem, or did they find one that is only partially consistent with the information given?

For example, if they only take into account $AC$, $BC$, and the angle $\angle A$, then they may obtain two possible values for the angle $\angle C$, but only one of them will be consistent with the known value of $AB$. If you ignore the value of $AB$, you get in trouble. So you should check if the "other value" they found for $C$ is consistent with the length of $AB$, and I'll bet you it is not.

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Yes. You're right. I don't get a consistent value for AB if I am using the value I got for $\angle C\$. – kuttaka Feb 3 '11 at 20:09

Could it be because when taking the inverse sine somebody forgot that there are two angles with the same sine, one less than 90 degrees and one greate?. In this case I get $\measuredangle B=119.6^\circ$ but you could think it is $61.4^\circ$

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It's ambiguous who this "somebody" is, but I think you're talking about me. – kuttaka Feb 3 '11 at 20:06