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.files.wordpress.com/2014/10/philistines.png

philistines

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.

SSA1

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.

SSA2

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}.

SSA4

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).

 

SSA3Many 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.

 

Leave a comment

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: