# The Law of Cosines
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/law-of-cosines/
Fetched from algebrica.org post 7152; source modified 2026-05-02T15:38:48.
## 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](../pythagorean-theorem.md), 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](../sine-and-cosine.md) 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](../pythagorean-identity.md) 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](../law-of-sines.md), 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](../vectors.md) 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\\).
Theorem
The structure of the entry is shown in the conceptual map, where each branch represents a core component and the sub-nodes highlight the specific notions discussed.
Intermediate
2
Requires
0
Enables
The following concepts, [Right Triangle Trigonometry](../right-triangle-trigonometry.md), [Sine and Cosine](../sine-and-cosine.md), are required as prerequisites for this entry.
