Cosine Rule: how to use it
The cosine rule connects the three sides of a triangle with one angle. It is especially useful when you know:
- Two sides and the included angle, and want the third side.
- All three sides, and want an angle.
Method for finding a side
- Label the side you want as the one opposite the known angle.
- Substitute into the cosine rule in the form that matches the known information.
- Calculate carefully, then simplify the final result if possible.
A common rearrangement is:
[ c^2 = a^2 + b^2 - 2ab\cos(C) ]
where (C) is the angle between sides (a) and (b).
Method for finding an angle
- Put the unknown angle in the formula and rearrange so that (\cos(\theta)) is alone.
- Evaluate the fraction first.
- Use the inverse cosine to find the angle.
Check your answer
- If finding a side, the answer should be positive and reasonable compared with the other sides.
- If finding an angle, check that it lies between (0^\circ) and (180^\circ).
- If the triangle is obtuse, the longest side should be opposite the largest angle.
Final tip
Match the labels carefully before substituting. Most errors come from using the wrong side opposite the angle.