The cosine law is a trigonometry law which can be used to derive the missing lengths of sides and angles of corners in triangles based on the lengths and angles of known sides. To do this, it requires knowledge of the lengths of at least 2 sides of the triangle, and either a known length of the third side, or the angle of the corner between the length of the known sides. Like with other trigonometric laws for non right triangles, this law defines the sides using the lowercase letters a, b and c and corners as uppercase letters A, B and C (Figure 1)

Figure 1: The corners of a triangle are labelled with capital letters A, B, and C. The opposite sides of these corners are labelled with the lower case letters a, b, and c. This allows for the cosine law to describe the sides and corners of a triangle using these letters to identify them.

There are two applications of the cosine law, and it is simpler to express these are different formulae. However these two equations are rearrangements of the same law, and the equation that defines them. These different equations can be used to determine the length of a missing side (c) (Eq. 1) or the angle of a missing corner (C) (Eq. 2), given the length of two other sides (a and b) and the side (c) or corner (C) opposite to the one being determined. In the cosine law equations, the lengths of sides a and b are interchangeable, as long as they are the two sides adjacent to the corner specified as C, while side c and corner C are always the side or corner of interest.

Eq. 1: \(c^2=a^2+b^2-2ab\cos C\)

Eq. 2: \(\cos C = {a^2+b^2-c^2 \over 2ab}\)

Cosine Law for a Missing Side

The cosine law can be used to find the length of a missing side if the lengths of the other two sides are known, along with the angle between them (Figure 2).

Figure 2: The cosine law can be used to find the length of a side opposite to a corner by using a known angle, and that corner’s two adjacent sides. The angle of the corner, and the lengths of the side can substitute into the equation, and as that side of the equation produces a value which is equal to the square of the length of the side being calculated, the square root of this result is the length of the side opposite the corner with the known angle.

To do this the lengths of the known sides and the angle of the known corner are substituted into the cosine law equation for the length of a side (Eq. 3). The length of the side being calculated is always defined as c, and the angle of the known corner opposite this side is always defined as C. The other two known sides are substituted into the equation as a and b. It doesn’t matter which side is substituted into the place of each letter, they are interchangeable and are treated the same by the equation.

Eq. 3: \(c^2=35^2+45^2-(2\times35\times45\times\cos85.9)\)

As the equation is for the square of the side c, taking the square root of both sides will find the length of side c (Eq. 4) and thus solve the length of the triangle using the cosine law.

Eq. 4: \(c=\sqrt{35^2+45^2-(2\times35\times45\times\cos85.9)}\)

Cosine Law for a Missing Corner

If the lengths of three sides of a triangle are known, the cosine law can be used to find the angle of any corner in the triangle (Figure 3).

Figure 3: The cosine law can be used to determine the angle of a corner in a triangle, if the length of the side opposite to the triangle are known, along with the lengths of the two sides adjacent to the corner of interest. These can put into the cosine law formula for a corner, which after solving the equation gives the cosine of the angle of interest. The inverse cosine (cos^-1) function can be applied to the value to derive the angle of interest. By using the lengths of different sides substituted into different variables in the equation, all the angles of the corner can be solved. Note that cosine law can be used to find missing angles if the length of all the sides are known, but not vice versa, as an infinite number of triangles are able to satisfy three known angles in a triangle if none of the sides are known.

In this case, the lengths of the sides in the triangle can be substituted into the cosine law equation (Eq. 5). The angle being solved is the angle of corner C, and the length of the side opposite this corner is defined as c. Like with the side law for the length of a side, the lengths of the other two sides are substituted as a and b, and these two values are interchangeable in the equation.

Eq. 5: \(\cos C={{35^2+45^2-55^2}\over{2\times35\times45}}\)

As the equation is for the cosine of angle C, we can apply the inverse cosine function, cos^-1 to both sides of the equation, to give the equation for the angle at corner C (Eq. 6).

Eq. 6: \(C=\cos^{-1} ({{35^2+45^2-55^2}\over{2\times35\times45}})=85.9\)