Sine rule, cosine rule and triangle area

With sides a, b and c opposite angles A, B and C, the sine rule is a/sin A = b/sin B = c/sin C. The cosine rule is c² = a² + b² − 2ab cos C. It uses two sides and their included angle, or three sides to find an angle. The triangle area is (1/2)ab sin C when C is the angle between sides a and b.

When using the sine rule to find an angle, an acute calculator result can have an obtuse supplement. Check whether each candidate produces a positive third angle and satisfies the other information. All triangle angles lie strictly between 0° and 180°. Use degree mode for these courses and retain enough intermediate precision to support the final rounding.

Worked example

Two sides of a triangle are 5 cm and 8 cm, with included angle 60°. Find the third side and the area.

1. c² = 25 + 64 − 2 × 5 × 8 × cos 60° = 49.
2. Therefore c = 7 cm.
3. Area = (1/2) × 5 × 8 × sin 60° = 20 × √3/2.

Answer: Third side 7 cm; area 10√3 cm²

Revise first: Exact trigonometric values.