For any triangle with side lengths a, b, c and opposite angles α, β, γ respectively, the law of cosines relates a side to the other two and the cosine of the included angle: c^2 = a^2 + b^2 − 2ab cos(γ).

Principle

It generalizes the Pythagorean theorem to non‑right triangles by introducing a correction term −2ab cos(γ) that accounts for the angle between sides; it encodes the inner‑product relation of vectors representing sides.

Demonstration

Given a triangle with sides a = 7, b = 5 and included angle γ = 60°, compute c^2 = 7^2 + 5^2 − 2·7·5·cos60° = 49 + 25 − 70·(1/2) = 74 − 35 = 39, so c = sqrt(39).

Misapplication

Using the formula with the wrong included angle (e.g., substituting an exterior angle) or sign errors in the cos term, or applying it verbatim in spherical or hyperbolic geometry where cosine laws have different forms.

Consequence

It allows solving any triangle from two sides and the included angle (SAS), classifying triangles by comparing c^2 to a^2 + b^2, and deriving vector length relations via dot products.

Reversal

When γ = 90°, cos(γ) = 0 and the law reduces to the Pythagorean theorem. If γ > 90° then cos(γ) < 0 and c^2 > a^2 + b^2, reflecting obtuse geometry.

Boundary

Valid in Euclidean geometry for planar triangles; on curved surfaces the corresponding spherical or hyperbolic cosine laws must be used. It presumes the standard Euclidean notion of angle and side length.

Semantic Tension

Relates to the vector dot‑product formulation r·s = |r||s|cosθ; tension arises when choosing between coordinate/vector methods and classical synthetic usages, though they are consistent in Euclidean space.

Synthesis

The law of cosines is the Euclidean identity that extends the Pythagorean relation to arbitrary triangles by incorporating the cosine of the included angle; it unifies side–angle–side computations and vector length relations.