Law of Cosines — When to Use It, Formula & Examples

The Law of Cosines generalizes the Pythagorean theorem to all triangles. When the angle is 90°, cos(90°) = 0 and it reduces to a² + b² = c². Use it when you have SAS or SSS — situations where the Law of Sines does not apply directly.

Law of Cosines (c²)

c² = a² + b² − 2ab·cos(C)

Step-by-Step Method

Identify What You Know

Law of Cosines requires SAS (two sides and the included angle) or SSS (all three sides). For SAS: you need sides a, b and the angle C between them. For SSS: you have all three sides and need to find an angle.

SAS or SSS → Law of Cosines

SAS Example: Find the Missing Side

Given a = 8, b = 6, C = 60°. Apply: c² = 8² + 6² − 2(8)(6)cos(60°) = 64 + 36 − 96(0.5) = 100 − 48 = 52. So c = √52 ≈ 7.21.

c² = a² + b² − 2ab·cos(C)

SSS Example: Find a Missing Angle

Given a = 5, b = 7, c = 8. Rearrange: cos(C) = (a² + b² − c²) / (2ab) = (25 + 49 − 64) / (2·5·7) = 10/70 ≈ 0.143. C = arccos(0.143) ≈ 81.8°.

cos(C) = (a² + b² − c²) / (2ab)

Verify with Triangle Properties

All angles must be positive and sum to 180°. All sides must satisfy the triangle inequality. If cos(C) = −1 to 1, the triangle is valid; outside this range means no valid triangle exists.

A + B + C = 180°

Law of Cosines vs Law of Sines

Use Law of Cosines for SAS and SSS cases. Use Law of Sines for ASA, AAS, and the ambiguous SSA case. Quick rule: if you have two sides and the angle BETWEEN them, use Cosines. If you have an angle-side pair across from each other, use Sines.

Real-World Applications

The Law of Cosines is used in surveying (calculating distances when direct measurement is impossible), navigation (finding distances between GPS coordinates), physics (vector addition), and architecture (roof angles, structural triangulation).

Frequently Asked Questions

When should I use Law of Cosines vs Law of Sines?▼

Use Cosines for SAS (two sides + included angle) or SSS (all three sides). Use Sines for AAS, ASA, or SSA (angle-side pair). Cosines is more versatile but involves more computation.

Is the Law of Cosines the same as the Pythagorean theorem?▼

The Pythagorean theorem is a special case of the Law of Cosines where angle C = 90°. Since cos(90°) = 0, the −2ab·cos(C) term drops out, leaving c² = a² + b².

Can the Law of Cosines give a negative result?▼

c² can never be negative (it represents a squared length), but cos(C) can be negative when C > 90°. A negative cosine means the angle is obtuse.

What if cos(C) is outside the range [−1, 1]?▼

If cos(C) falls outside [−1, 1] when solving for an angle from SSS, the three side lengths cannot form a valid triangle.

Try It Yourself

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