To determine the length of a side in a triangle using the cosine rule, you can apply the following formula:

The cosine rule is particularly advantageous for solving problems involving non-right-angled triangles. This formula establishes a relationship between the lengths of the sides of a triangle and the cosine of one of its angles. In the formula, $a$ represents the side you wish to calculate, while $b$ and $c$ are the lengths of the other two sides, and $A$ is the angle opposite side $a$.

To effectively use this formula, follow these steps:

1. Identify the sides and angle: Label the sides of the triangle as $a$, $b$, and $c$. Designate the angle opposite side $a$ as $A$.
2. Substitute known values: Insert the values of $b$, $c$, and $A$ into the formula.
3. Perform calculations step-by-step:
   • Begin by squaring the lengths of sides $b$ and $c$.
   • Next, calculate $2bc \cdot \cos(A)$.
4. Calculate the result: Subtract the result from the sum of the squares of $b$ and $c$.
5. Find the length of side $a$: Finally, take the square root of the result to determine the length of side $a$.

For example, let’s consider a scenario where $b = 5$, $c = 7$, and $A = 60^\circ$. We can apply the cosine rule as follows:

Calculating this step-by-step:

Taking the square root gives:

Thus, the length of side $a$ is approximately $6.24$ units.