Law of Sines
The Law of Sines
The Law of Sines is a topic in Right Triangles & 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 involving general triangles.
The Law of Sines states \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\), solving triangles from AAS, ASA, or the ambiguous SSA case.
Theory
The Law of Sines works in any triangle, not just right triangles:
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 part (AAS, ASA, or the ambiguous SSA).
The Law of Sines:
How to use the Law of Sines
- Match each side with its opposite angle.
- Set up a proportion with one complete pair and the unknown.
- Solve for the missing side or angle.
- Find the third angle first in ASA if needed.
Use \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}\).
| \(b\) | \(=\) | \(\dfrac{a\sin B}{\sin A}=\dfrac{10\sin 75^\circ}{\sin 40^\circ}\) |
| \(\approx\) | \(15.0\) |
First \(B=180^\circ-50^\circ-60^\circ=70^\circ\), then the Law of Sines.
| \(a\) | \(=\) | \(\dfrac{b\sin A}{\sin B}=\dfrac{8\sin 50^\circ}{\sin 70^\circ}\) |
| \(\approx\) | \(6.5\) |
Use \(\dfrac{\sin B}{b}=\dfrac{\sin A}{a}\).
| \(\sin B\) | \(=\) | \(\dfrac{9\sin 55^\circ}{12}\approx 0.614\) |
| \(B\) | \(\approx\) | \(37.9^\circ\) |
When you know an angle and its opposite side, plus one more piece — cases AAS, ASA, and the ambiguous SSA.
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: AAS, ASA, or SSA.
How do you find a side with the Law of Sines?
Use side-over-sine: \(b=\dfrac{a\sin B}{\sin A}\).
How do you find an angle with the Law of Sines?
Use sine-over-side: \(\sin B=\dfrac{b\sin A}{a}\), then take the inverse sine.