Choosing the right tool
Real problems combine the three results. Decide from what is given: use the sine rule when a side is paired with its opposite angle (AAS/ASA/SSA); use the cosine rule for two sides and the included angle (SAS) or all three sides (SSS); use ½ab sin C for area once two sides and their included angle are known.
Key formula
Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A. Area: ½ ab sin C. Angle sum: A + B + C = 180°.
Bearings and 3-D
Bearings are measured clockwise from north (000°–360°). Sketch the triangle, mark north lines, then apply the correct rule. In 3-D problems, identify the plane triangle you need and solve it separately.
Worked example
A ship sails 10 km from P to Q, turns and sails 24 km from Q to R, with angle PQR = 90°. Find PR and the area of triangle PQR. By the cosine rule (or Pythagoras since the angle is 90°): PR² = 10² + 24² − 2(10)(24)cos 90° = 100 + 576 − 0 = 676, so PR = 26 km. Area = ½ × 10 × 24 × sin 90° = 120 km².
A step-by-step strategy
Read the problem and draw a clear diagram. List every given side and angle, then decide which triangle contains the unknown. Match the data to a rule: a side with its opposite angle points to the sine rule; two sides and the angle between them, or three sides, point to the cosine rule; two sides and the included angle also give the area. In bearing problems, mark the north direction at each point and use angles on a straight line and around a point to build the interior angles of the triangle before applying a rule.
Remember
- Sketch first; label sides opposite their angles.
- Find the largest angle from the longest side to detect obtuse cases.
- Bearings run clockwise from north.