Law of Sines (with ambiguous case)
Law of Sines
Law of Sines is the opening topic of Triangle Trigonometry in the California Common Core State Standards. It is aligned to Standard G-SRT.11, which requires students to prove the Law of Sines and use it to solve problems.
The Law of Sines states \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) for any triangle, solving the AAS, ASA, and the ambiguous SSA cases.
Theory
The Law of Sines holds in any triangle, not just right triangles:
where each side is labeled with the lowercase letter of its opposite angle. Use it when you know an angle and its opposite side, plus one more piece:
- AAS or ASA — two angles and any side (always one triangle).
- SSA — two sides and a non-included angle (the ambiguous case: zero, one, or two triangles).
The Law of Sines, in both forms:
How to solve with the Law of Sines
- Match each side with its opposite angle.
- Set up a proportion using one complete pair and the unknown.
- Solve for the missing side or angle.
- For SSA, check the supplement of the angle you find — it may give a second valid triangle.
Use \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}\) and solve for \(b\).
| \(b\) | \(=\) | \(\dfrac{a\sin B}{\sin A}=\dfrac{10\sin 75^\circ}{\sin 40^\circ}\) |
| \(\approx\) | \(15.0\) |
First \(B=180^\circ-50^\circ-60^\circ=70^\circ\), then apply the Law of Sines.
| \(a\) | \(=\) | \(\dfrac{b\sin A}{\sin B}=\dfrac{8\sin 50^\circ}{\sin 70^\circ}\) |
| \(\approx\) | \(6.5\) |
Solve \(\dfrac{\sin B}{b}=\dfrac{\sin A}{a}\) for \(\sin B\).
| \(\sin B\) | \(=\) | \(\dfrac{b\sin A}{a}=\dfrac{9\sin 55^\circ}{12}\) |
| \(\approx\) | \(0.614\) | |
| \(B\) | \(\approx\) | \(37.9^\circ\) |
This is SSA: the side opposite the known angle is shorter than the other known side, so the swinging side \(a\) can meet the base at two points.
| \(\sin B\) | \(=\) | \(\dfrac{8\sin 35^\circ}{6}\approx 0.765\) |
| \(B\) | \(\approx\) | \(49.9^\circ\ \text{ or }\ 130.1^\circ\) |
Both angles are valid (they keep the angle sum under \(180^\circ\)), giving two triangles.
Common pitfalls
Frequently asked questions
What is the Law of Sines?
In any triangle, \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\); each side is proportional to the sine of its opposite angle.
When do you use the Law of Sines?
When you know an angle and its opposite side plus one more part: cases AAS, ASA, or SSA.
What is the ambiguous case?
The SSA situation, where the given data can produce zero, one, or two triangles. Always check whether a supplementary angle also fits.
How do you know if there are two triangles?
After finding an angle, test its supplement: if adding it to the known angle still leaves room under \(180^\circ\), a second triangle exists.