Arccosine has an important advantage over arcsine when solving for the parts of a triangle: there is no possibility ambiguity about the angle.

Solve if , , and .

When solving for the three angles, it’s best to start with the biggest angle (that is, the angle opposite the biggest side). To see why, let’s see what happens if we first use the Law of Cosines to solve for one of the two smaller angles, say :

So far, so good. Now let’s try using the Law of Sines to solve for :

Uh oh… there are two possible solutions for since, hypothetically, could be in either the first or second quadrant! So we have no way of knowing, using only the Law of Sines, whether or if .

For this reason, it would have been far better to solve for the biggest angle first. For the present example, the biggest answer is since that’s the angle opposite the longest side.

Using a calculator, we find that .

We now use the Law of Sines to solve for either or (pretending that we didn’t do the work above). Let’s solve for :

This equation also has two solutions in the interval , namely, and . However, we know full well that the answer can’t be larger than since that’s already known to be the largest angle. So there’s no need to overthink the matter — the answer from blindly using arcsine on a calculator is going to be the answer for .

Naturally, the easiest way of finding is by computing .

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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