The cosine rule, which is essential for determining an angle in any triangle (not just right-angled triangles), is expressed as:

This formula establishes a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. Specifically, if you have a triangle with sides $a$, $b$, and $c$, and you wish to find the angle $C$ that is opposite to side $c$, you can apply this formula.

To utilize the cosine rule, you must know the lengths of all three sides of the triangle. Once you have these measurements, you can substitute them into the formula. After calculating the right-hand side, you will obtain the cosine of angle $C$. To determine the angle itself, you can then apply the inverse cosine function, commonly denoted as $\cos^{-1}$ or $\arccos$, using your calculator.

For example, consider a triangle with sides $a = 5$, $b = 7$, and $c = 8$. You would substitute these values into the cosine rule as follows:

To find angle $C$, you would calculate:

The cosine rule is particularly advantageous when dealing with non-right-angled triangles, as it allows you to find angles where simpler trigonometric ratios—such as sine, cosine, or tangent—cannot be applied directly. Furthermore, it can also be used in reverse to calculate a side length if you know the lengths of two sides and the measure of the included angle.