Law of Sines: The Ambiguous Case

This article will teach you how to use the law of sines to solve ambiguous SSA (side-side-angle) cases. But before, let's start by recalling the law of sines. The law of sines applies to any triangle, equating the ratios of the sines of the angles to the lengths of the corresponding sides.

If you want to know more about the law of sines, check out our detailed article: What is the Law of Sines? (Sine Rule Explained).

In short, we can use the law of sines to find unknown angles and lengths in triangles where we know either two lengths and an opposite angle, or two angles and an opposite side.

Using the law of sines to determine an unknown side or angle can result in an ambiguous answer. The ambiguous case of the sine rule occurs under the following conditions:

• You only know the angle $\alpha$ and sides $a$ and $c$;
• The angle $α$ is acute ($α \lt 90^\circ$);
• $a$ is shorter than $c$ ($a \lt c$); and
• $a$ is longer than the altitude $h$ from angle $\beta$, where $h = c \times \sin \alpha$ (or $a \lt c \times \sin \alpha$).

This situation leads to two possible solutions because the same sine value corresponds to two different angles: $\theta$ and $180^\circ - \theta$.

It is important to note that the law of sines and the ambiguous case can only occur if we are given the lengths of two sides and an opposite angle; however, depending on the values, there are three possibilities to distinguish in this case: no triangle exists, one triangle exists, or two triangles exist.

We have summarized the possible cases when two sides $a$ and $b$, and an opposite angle $\alpha$ are known in the table below.

Suppose you have a triangle $ABC$ with $a = 3 \ \text{in}$, $b = 9 \ \text{in}$, $\alpha = 10^\circ$. Let's determine all possible measures of $\beta$ (to the nearest degree).

1. Check how many triangles are possible. Since $\alpha$ is acute, we calculate the altitude:

Because $h < a < b$, it is possible to build two different triangles.

2. Apply the law of sines 🇺🇸:

3. Solve for $\beta$:

4. Since $\sin(\theta)$ has the same value for $\theta$ and $180^\circ - \theta$, there is another possible solution:

5. Check if the triangle angle sum is less than $180^\circ$:

So this value of $\beta$ is also possible.

There are two possible measures for $\beta$: $\beta \approx 31.4^\circ$ or $\beta\approx 148.6^\circ$. Hence, two distinct triangles can be formed.

The ambiguous case of the sine rule occurs in the SSA configuration, when two sides and a non-included angle are given. In this situation, the law of sines may produce two triangles, one triangle, or no triangle at all. Knowing how to test for acute and obtuse angles allows you to interpret the results correctly.