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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 that lets you solve triangles with no right angle at all, which come up constantly in navigation, surveying, and construction.

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, unlike AAS and ASA which always resolve to exactly one triangle.

The formula works because the ratio of a side to the sine of its opposite angle is actually the diameter of the triangle's circumscribed circle — the same circle that passes through all three of the triangle's vertices. That's why the ratio is the same for all three side-angle pairs in a given triangle, even though it isn't obvious just from looking at the formula.

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)
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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

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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.

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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.

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Algebra
a² + b² = c²

Pythagorean Theorem

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

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