Inverse functions: Arcsine and SSA (Part 17)

In the last few posts, we studied the SSA case of solving for a triangle, when two sides and an non-included angle are given. (Some mathematics instructors happily prefer the angle-side-side acronym to bluntly describe the complications that arise from this possibly ambiguous case. I personally prefer not to use this acronym.)

A note on notation: when solving for the parts of $\triangle ABC$ , $a$ will be the length of the side opposite $\angle A$ , $b$ will be the length of the side opposite $\angle B$ , and $c$ will be the length of the side opposite angle $C$ . Also $\alpha$ will be the measure of $\angle A$ , $\beta$ will be measure of $\angle B$ , and $\gamma$ will be the measure of $\angle C$ . Modern textbooks tend not to use $\alpha$ , $\beta$ , and $\gamma$ for these kinds of problems, for which I have only one response:https://meangreenmath.com/wp-content/uploads/2014/10/philistines.png

Suppose that $a$ , $c$ , and the nonincluded angle $\alpha$ are given, and we are supposed to solve for $b$ , $\beta$ , and $\gamma$ . As we’ve seen in this series, there are four distinct cases — and handling these cases requires accurately solving equation like $\sin \gamma = \hbox{something}$ on the interval $[0^\circ, 180^\circ]$ .

Case 1. $b < c \sin \alpha$ . In this case, there are no solutions. When the Law of Sines is employed and we reach the step

$\sin \gamma = \hbox{something}$

the $\hbox{something}$ is greater than 1, which is impossible.

Case 2. $b = c \sin \alpha$ . This rarely arises in practice (except by careful writers of textbooks). In this case, there is exactly one solution. When the Law of Sines is employed, we obtain

$\sin \gamma = 1$

We conclude that $\gamma = 90^\circ$ , so that $\triangle ABC$ is a right triangle.

Case 3. $c \sin \alpha < b < c$ . This is the ambiguous case that yields two solutions. The Law of Sines yields

$\sin \gamma = \hbox{something}$

so that there are two possible choices for $\gamma$ , $\hbox{some angle}$ and $180^\circ - \hbox{some angle}$ .

Case 4. $b > c$ . This yields one solution. Similar to Case 3, the Law of Sines yields

$\sin \gamma = \hbox{something}$

so that there are two possible choices for $\gamma$ , $\hbox{some angle}$ and $180^\circ - \hbox{some angle}$ . However, when the second larger value of $\gamma$ is attempted, we end up with a negative angle for $\beta$ , which is impossible (unlike Case 3).

SSA3 Many mathematics students prefer to memorize rules like those listed above. However, I try to encourage my students not to blindly use rules when solving the SSA case, as it’s just too easy to make a mistake in identifying the proper case. Instead, I encourage them to use the Law of Sines and to remember that the equation

$\sin \gamma = t$

has two solutions in $[0^\circ, 180^\circ]$ as long as $0 < t < 1$:

$\gamma = \sin^{-1} t \qquad \hbox{and} \qquad \gamma = 180^\circ - \sin^{-1} t$

If they can remember this fact, then students can just follow their noses when applying the Law of Sines, identifying impossible and ambiguous cases when the occasions arise.

Inverse functions: Arcsine and SSA (Part 17)

Published by John Quintanilla

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