Inverse Functions: Arccosine and SSS (Part 19)

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

Solve \triangle ABC if latex a = 16$, b = 20, and c = 25.

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


c^2 = a^2 + b^2 - 2 a b \cos \gamma

625 = 256 + 400 - 640 \cos \gamma

-31 =-640 \cos \gamma

0.0484375 = \cos \gamma

Using a calculator, we find that \gamma \approx 87.2^\circ. And the good news is that there is no need to overthink this… this is guaranteed to be the angle since the range of y = \cos^{-1} x is [0,\pi], or [0^\circ, 180^\circ] in degrees. So the equation

\cos x = \hbox{something}

is guaranteed to have a unique solution between 0^\circ and 180^\circ. (But there are infinitely many solutions on \mathbb{R}. And since an angle in a triangle must lie between 0^\circ and 180^\circ, 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 [0^\circ,90^\circ] and also the interval [90^\circ, 180^\circ].

From this point forward, the Law of Cosines could be employed again to find either \alpha or \beta. Indeed, this would be my preference since the sides a, b, and c 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 c (exactly known) and \gamma (approximately known with a calculator).

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