Cosine Law

trigonometry

Definition

Cosine Law

The Cosine Law is a generalisation of the Pythagorean theorem that relates the lengths of the sides of a triangle to the cosine of one of its interior angles. For a triangle with sides $a,b,c$ and corresponding opposite angles $\alpha ,\beta ,\gamma$, the law provides three equivalent identities:

$$\begin{array}{rl}{a}^2& =b^2+c^2-2bc\cos (\alpha )\\ b^2& =a^2+c^2-2ac\cos (\beta )\\ c^2& =a^2+b^2-2ab\cos (\gamma )\end{array}$$

Structural Symmetry

The formula exhibits cyclic symmetry across the vertices of the triangle. Each side can be expressed as the square root of the sum of the squares of the remaining two sides minus twice their product scaled by the cosine of the included angle. This pattern ensures that the relationship holds regardless of the chosen orientation or labelling of the triangle’s components.

Generalisation of Pythagoras

The Cosine Law encompasses the Pythagorean theorem as a special case. When the angle $\gamma$ is a right angle ($\gamma =\frac{\pi }{2}$ or $90^{\circ }$), the term $\cos (\gamma )$ becomes zero, and the equation simplifies to $c^2=a^2+b^2$. This demonstrates that the law provides the necessary correction for non-orthogonal triangle configurations.

Vector Interpretation

In the context of vector algebra, the sides of the triangle can be represented as vectors $\mathbf{a}$ and $\mathbf{b}$ meeting at an origin $C$. The third side is then given by the vector difference $\mathbf{c}=\mathbf{a}-\mathbf{b}$. The square of the length of $\mathbf{c}$ is derived from the dot product:

$\Vert \mathbf{c}{\Vert }^2=(\mathbf{a}-\mathbf{b})\cdot (\mathbf{a}-\mathbf{b})=\Vert \mathbf{a}{\Vert }^2+\Vert \mathbf{b}{\Vert }^2-2(\mathbf{a}\cdot \mathbf{b})$

Substituting the geometric definition of the dot product:

$$\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos (\gamma )$$

yields the standard scalar form of the law.