cosine rule

The cosine rule is used to discover the length of a missing side or the size of a missing angle in a triangle. It can be a right-angled triangle or a triangle with no right angles.

The rule is based on the relationship between two sides of a triangle and the angle between those sides.

The rule has two forms:

(1) To discover the length of a side:

    \[ a^2 = b^2 + c^2 - 2bc\cos A \\ \]

    \[ b^2 = a^2 + c^2 - 2ac\cos B \\ \]

    \[ c^2 = a^2 + b^2 - 2ab\cos C \]

(2) To discover the size of an angle:

    \[ cos A = \frac{b^2 + c^2 -a^2}{2bc} \]

    \[ cos B = \frac{a^2 + c^2 -b^2}{2ac} \]

    \[ cos C = \frac{a^2 + b^2 -c^2}{2ab} \]

The letters a, b, c refer to the sides of the triangle, and the letters A, B, C refer to the angles of the triangle that are opposite the sides a, b, c.