Trigonometry · SS 1, SS 2
Cosine rule
Find the third side from two sides and the angle between them, or an angle from three sides.
- a2 = b2 + c2 − 2bc cos A
- cos A = b2 + c2 − a22bc
What the letters mean
- a
- the side opposite angle A
- b, c
- the two sides that make angle A
Worked example
Two sides of a triangle are 5 cm and 8 cm with 60° between them. Find the third side.
Step 1: Put in the numbers
a2 = 52 + 82 − 2 × 5 × 8 × cos 60°
Step 2: cos 60° = 0.5
a2 = 25 + 64 − 40 = 49
Step 3: Square root
a = 7 cm
Final answer: 7 cm
When A = 90°, cos A = 0 and the cosine rule becomes Pythagoras.
See it
Play with this picture from the Bearings and Distances lesson.
Bearing of B from A
065°
Back bearing (A from B)
245°
065° + 180° = 245°
B is 2.54 km north and 5.44 km east of A.
Distance A to B6 km