Trigonometry — Cosine rule

About this lesson

This unit focuses on the cosine rule for any triangle. It begins by motivating the rule through a discussion of triangles where two sides and a non-included angle are known, showing that the sine rule is insufficient in such cases. The cosine rule is then derived using the Pythagorean theorem and presented in three forms, one for each side of a triangle. The unit explains how to use the cosine rule to find an unknown side when two sides and the included angle are given, and to find an unknown angle when all three sides are given. It also covers solving a triangle completely, either from two sides and the included angle or from three sides. Worked examples include finding a side, finding an angle, finding the largest angle in a triangle, and proving a quadrilateral is cyclic by showing opposite angles are supplementary. Exercises reinforce these skills and include problems involving quadrilaterals and parallelograms. By the end of the unit, a student should be able to apply the cosine rule to find unknown sides and angles in triangles, solve triangles from given data, and use the rule in geometric proofs involving cyclic quadrilaterals.