The Law of Cosines

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Source: algebrica.org — CC BY-NC 4.0 https://algebrica.org/law-of-cosines/

Definition

The law of cosines relates the sides of any triangle through the angle opposite to one of them. It can be viewed as a generalisation of the Pythagorean theorem, valid not only for right triangles but for every triangle: the square of a side equals the sum of the squares of the other two sides, minus a corrective term that accounts for how open the angle between them is. For a triangle with sides $a, b, c$ and angle $\theta$ opposite to side $c$, the law states:

c^2 = a^2 + b^2 - 2ab \cos(\theta)

When $\theta = 90^\circ$ the cosine term vanishes and the formula reduces exactly to the Pythagorean theorem, which confirms that the law of cosines is a strict generalisation of that result. For any other angle, the corrective term either subtracts from or adds to the sum $a^2 + b^2$, depending on whether $\theta$ is acute or obtuse.

To derive the formula, drop the altitude $h$ from the vertex opposite to $c$ to the side $b$. This divides $b$ into two segments: $m = a\cos(\theta)$ and $n = b - a\cos(\theta)$, while the altitude itself satisfies $h = a\sin(\theta)$. Applying the Pythagorean theorem to the right triangle formed by $n$, $h$ and $c$ gives:

\begin{align} c^2 &= n^2 + h^2 \\[6pt] &= (b - a\cos(\theta))^2 + (a\sin(\theta))^2 \\[6pt] &= b^2 - 2ab\cos(\theta) + a^2\cos^2(\theta) + a^2\sin^2(\theta) \\[6pt] &= b^2 - 2ab\cos(\theta) + a^2(\cos^2(\theta) + \sin^2(\theta)) \end{align}

Since the Pythagorean identity gives $\sin^2(\theta) + \cos^2(\theta) = 1$, the expression simplifies to:

c^2 = a^2 + b^2 - 2ab\cos(\theta)

The law of cosines is often used in conjunction with the law of sines, which provides a complementary approach to solving triangles when different combinations of sides and angles are known.

Example 1

Consider a triangle with sides $a = 8$, $b = 6$ and included angle $\theta = 60^\circ$. The goal is to determine the length of the third side $c$. Substituting the known values into the law of cosines gives:

\begin{align} c^2 &= a^2 + b^2 - 2ab\cos(\theta) \\[6pt] &= 64 + 36 - 2(8)(6)\cos(60^\circ) \\[6pt] &= 64 + 36 - 96 \cdot \frac{1}{2} \\[6pt] &= 100 - 48 \\[6pt] &= 52 \end{align}

Taking the positive square root, one obtains $c = \sqrt{52} = 2\sqrt{13} \approx 7.21$.

The length of the third side is approximately $7.21$ units.

Example 2

Consider a triangle with sides $a = 5$, $b = 7$ and $c = 9$. The goal is to determine the angle $\theta$ opposite to side $c$. Solving the law of cosines for $\cos(\theta)$ gives:

\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}

Substituting the known values:

\begin{align} \cos(\theta) &= \frac{25 + 49 - 81}{2(5)(7)} \\[6pt] &= \frac{-7}{70} \\[6pt] &= -0.1 \end{align}

Since $\cos(\theta) < 0$, the angle $\theta$ is obtuse. Taking the inverse cosine yields:

\theta = \arccos(-0.1) \approx 95.7^\circ

The angle opposite to the longest side is approximately $95.7^\circ$.

Vector interpretation

The law of cosines admits a reading in terms of vectors that exposes its deeper structure and connects it to the inner product. Consider a triangle with vertex $O$, and let $\vec{u}$ and $\vec{v}$ denote the two sides of length $a$ and $b$ issuing from $O$, so that $a = \|\vec{u}\|$ and $b = \|\vec{v}\|$. The third side of the triangle, of length $c$, is then represented by the vector $\vec{v} - \vec{u}$, which joins the endpoints of $\vec{u}$ and $\vec{v}$. Expanding the squared norm of this vector through the bilinearity of the inner product gives:

\begin{align} \|\vec{v} - \vec{u}\|^2 &= (\vec{v} - \vec{u}) \cdot (\vec{v} - \vec{u}) \\[6pt] &= \|\vec{v}\|^2 - 2\,\vec{u} \cdot \vec{v} + \|\vec{u}\|^2 \end{align}

The geometric definition of the inner product states that:

\vec{u} \cdot \vec{v} = \|\vec{u}\|\|\vec{v}\|\cos\theta

$\theta$ is the angle between the two vectors at $O$, which coincides with the angle between the sides $a$ and $b$ of the triangle. Substituting this identity into the expansion above gives:

c^2 = a^2 + b^2 - 2ab\cos\theta

From this point of view the law of cosines is a reformulation of the identity that defines the inner product in terms of lengths and angles. The corrective term $-2ab\cos\theta$ that distinguishes a generic triangle from a right one is nothing other than $-2\,\vec{u} \cdot \vec{v}$, and the Pythagorean case corresponds to the situation in which the two vectors are orthogonal, so that $\vec{u} \cdot \vec{v} = 0$.