Law of Sines

📖 What is the Law of Sines?

The law of sines relates the sides of any triangle to the sines of their opposite angles. Unlike SOH-CAH-TOA (which only works for right triangles), the law of sines works for all triangles.

a/sin(A) = b/sin(B) = c/sin(C)

🔤 What Each Variable Means

a, b, c

Side lengthsThe three sides of the triangle

A, B, C

Opposite anglesEach angle is opposite its corresponding lowercase side (A is opposite a, etc.)

📝 Step-by-Step Example

In a triangle, A = 30°, B = 70°, and side a = 8. Find side b.

1

Set up the ratioa/sin(A) = b/sin(B)

2

Substitute known values8/sin(30°) = b/sin(70°)

3

sin(30°) = 0.5 and sin(70°) ≈ 0.93978/0.5 = b/0.9397 → 16 = b/0.9397

✓

Answer: b = 16 × 0.9397 ≈ 15.04

💡 When to Use the Law of Sines

AAS — two angles and one side known

ASA — two angles and the included side known

SSA — two sides and a non-included angle (ambiguous case — may have 0, 1, or 2 solutions)

📖 What is the Law of Sines?

🔤 What Each Variable Means

📝 Step-by-Step Example

💡 When to Use the Law of Sines

🔗 Related