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

Solve latex a = 16$, , and .

To solve for, say, the angle , we employ the Law of Cosines:

Using a calculator, we find that . And the good news is that there is no need to overthink this… this is guaranteed to be the angle since the range of is , or in degrees. So the equation

is guaranteed to have a unique solution between and . (But there are infinitely many solutions on . And since an angle in a triangle must lie between and , the practical upshot is that just plugging into a calculator blindly is perfectly OK for this problem. This is in stark contrast to the Law of Sines, for which some attention must be paid for solutions in the interval and also the interval .

From this point forward, the Law of Cosines could be employed again to find either or . Indeed, this would be my preference since the sides , , and are exactly. However, my experience is that students prefer the simplicity of the Law of Sines to solve for one of these angles, using the now known pair of (exactly known) and (approximately known with a calculator).

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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