The 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:
\[c^{2} = a^{2} + b^{2} - 2 a b cos ( \theta )\]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:
\[c^{2} & = n^{2} + h^{2} \\ & = ( b - a cos ( \theta ) )^{2} + ( a sin ( \theta ) )^{2} \\ & = b^{2} - 2 a b cos ( \theta ) + a^{2} cos^{2} ( \theta ) + a^{2} sin^{2} ( \theta ) \\ & = b^{2} - 2 a b cos ( \theta ) + a^{2} ( cos^{2} ( \theta ) + sin^{2} ( \theta ) )\]Since the Pythagorean identity gives $sin^{2} ( \theta ) + cos^{2} ( \theta ) = 1$, the expression simplifies to:
\[c^{2} = a^{2} + b^{2} - 2 a b cos ( \theta )\]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:
\[c^{2} & = a^{2} + b^{2} - 2 a b cos ( \theta ) \\ & = 64 + 36 - 2 ( 8 ) ( 6 ) cos ( 60^{\circ} ) \\ & = 64 + 36 - 96 \cdot \frac{1}{2} \\ & = 100 - 48 \\ & = 52\]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:
\[cos ( \theta ) = \frac{a^{2} + b^{2} - c^{2}}{2 a b}\]Substituting the known values:
\[cos ( \theta ) & = \frac{25 + 49 - 81}{2 ( 5 ) ( 7 )} \\ & = \frac{- 7}{70} \\ & = - 0.1\]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 $\overset{\rightarrow}{u}$ and $\overset{\rightarrow}{v}$ denote the two sides of length $a$ and $b$ issuing from $O$, so that $a = | \overset{\rightarrow}{u} | $ and $b = | \overset{\rightarrow}{v} | $. The third side of the triangle, of length $c$, is then represented by the vector $\overset{\rightarrow}{v} - \overset{\rightarrow}{u}$, which joins the endpoints of $\overset{\rightarrow}{u}$ and $\overset{\rightarrow}{v}$. Expanding the squared norm of this vector through the bilinearity of the inner product gives: |
The geometric definition of the inner product states that:
\[\overset{\rightarrow}{u} \cdot \overset{\rightarrow}{v} = | \overset{\rightarrow}{u} | | \overset{\rightarrow}{v} | cos \theta\]$\theta$ 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:
\[c^{2} = a^{2} + b^{2} - 2 a b cos \theta\]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 $- 2 a b cos \theta$ that distinguishes a generic triangle from a right one is nothing other than $- 2 \overset{\rightarrow}{u} \cdot \overset{\rightarrow}{v}$, and the Pythagorean case corresponds to the situation in which the two vectors are orthogonal, so that $\overset{\rightarrow}{u} \cdot \overset{\rightarrow}{v} = 0$.
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