How do you calculate a side using the cosine rule?

To calculate a side using the cosine rule, use the formula: \( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \).

The cosine rule is particularly useful in non-right-angled triangles. It relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is: \( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \), where \( a \) is the side you want to find, \( b \) and \( c \) are the other two sides, and \( A \) is the angle opposite side \( a \).

To use this formula, follow these steps:
1. Identify the sides and the angle in your triangle. Label the sides as \( a \), \( b \), and \( c \), and the angle opposite side \( a \) as \( A \).
2. Substitute the known values into the formula. For example, if you know \( b \), \( c \), and \( A \), plug these values into the equation.
3. Perform the calculations step-by-step. First, square the lengths of sides \( b \) and \( c \). Then, multiply \( b \) and \( c \) by 2 and by the cosine of angle \( A \).
4. Subtract the product from the sum of the squares of \( b \) and \( c \).
5. Finally, take the square root of the result to find the length of side \( a \).

For example, if \( b = 5 \), \( c = 7 \), and \( A = 60^\circ \):
\[ a^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60^\circ) \]
\[ a^2 = 25 + 49 - 70 \cdot 0.5 \]
\[ a^2 = 74 - 35 \]
\[ a^2 = 39 \]
\[ a = \sqrt{39} \approx 6.24 \]

So, the length of side \( a \) is approximately 6.24 units.