Inverse functions: Arcsine and SSA (Part 12)

We’ve seen in this series that blinding using the arcsine function on a calculator is insufficient for finding all solutions of an equation like \sin \theta = 0.8. In today’s post, I discuss one of the first places that this becomes practically important: solving the ambiguous case of solving a triangle given two sides and an nonincluded angle.

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:

philistines

Why does an SSA triangle produce an ambiguous case (unlike the SAS, SSS, or ASA cases)? Here’s a possible problem that has no solution:

Solve \triangle ABC if a = 3, c = 10, and \alpha = 30^\circ.

A student new to the Law of Sines might naively start solving the problem by drawing something like this:

badSSAOf course, that’s an inaccurate picture that isn’t drawn to scale. A more accurate picture would look like this:

SSA1The red dashed circle with center B illustrates the dilemma: “side” BC is simply too short to reach the horizontal dashed line to make the vertex C, dangling limply from the vertex B.

Of course, students should not be expected to make a picture this accurately when doing homework. Fortunately, this impossibility naturally falls out of the equation when using the Law of Sines:

\displaystyle \frac{\sin \alpha}{a} = \displaystyle \frac{\sin \gamma}{c}

\displaystyle \frac{\sin 30^\circ}{3} = \displaystyle \frac{\sin \gamma}{10}

\displaystyle \frac{1/2}{3} = \displaystyle \frac{\sin \gamma}{10}

\displaystyle \frac{5}{3} = \sin \gamma

Since \sin \gamma must like between 0 and 1 (said another way, \sin^{-1} \frac{5}{3} is undefined), we know that this triangle cannot be solved.

green lineIn the next few posts of this series, I’ll consider the other SSA cases — including the case where two solutions are possible.

 

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