20.7 The cosine rule — Learn

Prerequisites: cosine, square roots, and rearranging equations.

Objectives: use the cosine rule to find a side or angle in a non-right-angled triangle.

The cosine rule finds a side from two sides and their included angle: $a^2=b^2+c^2-2bc\cos A$. It can also find an angle by rearranging the formula.

Example 1. Two sides are $5$ cm and $7$ cm, with included angle $60^\circ$. Find the opposite side $a$.

Example 2. Find angle $A$ when the opposite side is $8$ cm and the other sides are $5$ cm and $6$ cm.

Watch out. The angle in the formula must sit between the two known sides. Check the diagram before substituting values.

Progress check

The garden diagonal uses Pythagoras because its triangle is right-angled. In a triangle without a right angle, choose a suitable sine or cosine rule.

You should now be able to

• use the included angle and two adjacent sides to find a third side;
• rearrange the cosine rule to find an angle;
• check that the chosen angle matches the sides in the formula.