Inverse functions: Arcsine and SSA (Part 14)

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:

Why does an SSA triangle produce an ambiguous case (unlike the SAS, SSS, or ASA cases)? Here’s a possible problem that has exactly one solution:

Solve $\triangle ABC$ if $a = 15$, $c = 10$, and $\alpha = 30^\circ$.

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

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

Notice that the red circle intersects the dashed black line at exactly one point. Therefore, we know that there will be exactly one solution for this case. We also note that the circle would have intersected the black dashed line had the dashed line been extended to the left. This will become algebraically clear in the solution below.

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:

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

$\displaystyle \frac{\sin 30^\circ}{15} = \displaystyle \frac{\sin \gamma}{10}$

$\displaystyle \frac{1/2}{15} = \displaystyle \frac{\sin \gamma}{10}$

$\displaystyle \frac{1}{3} = \sin \gamma$

At this point, the natural inclination of a student is to pop out the calculator and find $\sin^{-1} \frac{1}{3}$.

This is incorrect logic that, as we’ll see tomorrow, nevertheless leads to the correct conclusion. This is incorrect logic because there are two angles between $0^\circ$ and $180^\circ$ with a sine of $1/3$. There is one solution in the first quadrant (the unique answer specified by arcsine), and there is another answer in the second quadrant — which is between $90^\circ$ and $180^\circ$ and hence not a permissible value of arcsine. Let me demonstrate this in three different ways.

First, let’s look at the graph of $y = \sin x$ (where, for convenience, the units of the $x-$axis are in degrees). This graph intersects the line $y = \frac{1}{3}$ in two different places between $0^\circ$ and $180^\circ$. This does not violate the way that arcsine was defined — arcsine was defined using the restricted domain $[-\pi/2,\pi/2]$, or $[-90^\circ, 90^\circ]$ in degrees.

Second, let’s look at drawing angles in standard position. The angle in the second quadrant is clearly the reflection of the angle in the first quadrant through the $y-$axis.

Third, let’s use a trigonometric identity to calculate $\sin \left( \pi - \sin^{-1} \displaystyle \frac{1}{3} \right)$:

$\sin \left( \pi - \sin^{-1} \displaystyle \frac{1}{3} \right) = \sin \pi \cos \left( \sin^{-1} \displaystyle \frac{1}{3} \right) - \cos \pi \sin \left( \sin^{-1} \displaystyle \frac{1}{3} \right)$

$=0 \cdot \cos \left( \sin^{-1} \displaystyle \frac{1}{3} \right) + 1 \cdot \sin \left( \sin^{-1} \displaystyle \frac{1}{3} \right)$

$= \displaystyle \frac{1}{3}$

Fourth, and perhaps most convincingly for modern students (to my great frustration), let’s use a calculator:

All this to say, blinding computing $\sin^{-1} \frac{1}{3}$ uses incorrect logic when solving this problem.

Tomorrow, we’ll examine what happens when we try to solve the triangle using these two different solutions for $\gamma$.