What is the cosine rule for finding a side?

The cosine rule for finding a side states: \( c^2 = a^2 + b^2 - 2ab \cos(C) \).

The cosine rule is a useful formula in trigonometry for finding a side of any triangle, not just right-angled ones. It relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, if you have a triangle with sides \(a\), \(b\), and \(c\), and the angle opposite side \(c\) is \(C\), then the formula is \( c^2 = a^2 + b^2 - 2ab \cos(C) \). This allows you to calculate the length of side \(c\) if you know the lengths of sides \(a\) and \(b\) and the measure of angle \(C\).

To use the cosine rule, follow these steps:
1. Identify the sides and the angle you are working with. Make sure you know the lengths of two sides and the measure of the included angle (the angle between the two known sides).
2. Substitute the known values into the formula \( c^2 = a^2 + b^2 - 2ab \cos(C) \).
3. Calculate the value of \( c^2 \) by performing the operations on the right-hand side of the equation.
4. Finally, take the square root of the result to find the length of side \(c\).

For example, if you have a triangle with sides \(a = 5\) cm, \(b = 7\) cm, and the included angle \(C = 60^\circ\), you would substitute these values into the formula to get \( c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60^\circ) \). Simplifying this, you find \( c^2 = 25 + 49 - 70 \cdot 0.5 = 25 + 49 - 35 = 39 \). Therefore, \( c = \sqrt{39} \approx 6.24 \) cm.