☰
🏠 Home📐 Mathematics⚡ Physics🧪 Chemistry📏 Geometryℹ️ About✉️ Contact
Trigonometry⏱ 5 min read

Law of Sines

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

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 every triangle — a genuine generalization.

It's most directly useful when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA) — though that last case is called the "ambiguous case" because it can have zero, one, or two valid solutions depending on the specific values.

What Each Variable Means

a, b, c
Side lengthsThe three sides of the triangle.
A, B, C
Opposite anglesEach angle is opposite its matching lowercase side — A is opposite a, and so on.

When to Use It

  • AAS — two angles and one side are known
  • ASA — two angles and the included side are known
  • SSA — two sides and a non-included angle are known (the ambiguous case — check for 0, 1, or 2 valid solutions)
Advertisement

Step-by-Step Example

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

1
Set up the ratio

Match the known angle-side pair to the unknown one.

a/sin(A) = b/sin(B)
2
Substitute known values

Plug in a, A, and B.

8/sin(30°) = b/sin(70°)
3
Compute the known ratio

sin(30°) = 0.5 and sin(70°) ≈ 0.9397.

8/0.5 = 16 = b/0.9397
✓
Answer: b = 16 × 0.9397 ≈ 15.04

Interactive Calculator

Result will appear here

Common Mistakes

  • Mistake: Applying the Law of Sines to a SAS or SSS triangle.

    Fix: When you know two sides and the included angle (SAS), or all three sides (SSS), use the Law of Cosines instead — the Law of Sines needs at least one angle-side opposite pair to set up the ratio.

  • Mistake: Missing the ambiguous case in SSA problems.

    Fix: Given two sides and a non-included angle, there can be two valid triangles, not just one — always check whether a second solution exists before finalizing an answer.

Practice Questions

  1. In a triangle, A = 40°, C = 60°, and side a = 10. Find side c.

    Hint: c/sin(C) = a/sin(A).

  2. In a triangle, A = 50°, B = 60°, and side b = 12. Find side a.

Frequently Asked Questions

Why is SSA called the "ambiguous case"?

Because knowing two sides and a non-included angle doesn't always pin down a unique triangle — depending on the values, there can be zero, one, or two different triangles that fit, unlike SAS or ASA which always give exactly one.

Can the Law of Sines find a missing angle instead of a side?

Yes — rearrange the same ratio to solve for sin of the unknown angle, then take the inverse sine. Just watch for the ambiguous case there too.

Related Formulas

Trigonometry
c² = a² + b² − 2ab·cos(C)

Law of Cosines

A generalization of the Pythagorean theorem that works for any triangle, not just right triangles.

Learn more →
Trigonometry
sin θ = O/H, cos θ = A/H, tan θ = O/A

Sine, Cosine & Tangent

The three fundamental trigonometric ratios for right triangles, remembered with SOH-CAH-TOA.

Learn more →
Algebra
a² + b² = c²

Pythagorean Theorem

Relates the three sides of a right triangle, letting you find any one side from the other two.

Learn more →