How do you use the cosine rule to find a missing angle?

To find a missing angle using the cosine rule, rearrange the formula to solve for the angle.

The cosine rule is typically written as \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( a \), \( b \), and \( c \) are the lengths of the sides of a triangle, and \( C \) is the angle opposite side \( c \). To find a missing angle, you need to rearrange this formula to solve for \( \cos(C) \).

First, isolate \( \cos(C) \) by moving the other terms to the other side of the equation:
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]

Next, substitute the known values of the sides \( a \), \( b \), and \( c \) into the equation. For example, if you know the lengths of all three sides of the triangle, plug these values into the formula:
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]

Once you have calculated the value of \( \cos(C) \), use the inverse cosine function (also known as arccos) on your calculator to find the angle \( C \):
\[ C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \]

Make sure your calculator is set to the correct mode (degrees or radians) depending on the context of your problem. This method allows you to find the measure of the missing angle accurately.