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The cosine rule

The cosine rule extends Pythagoras to every triangle. It is used when you know two sides and the angle between them, or all three sides. If the angle is $90^circ$ , $cos A = 0$ and the formula becomes Pythagoras' theorem.

a^2 = b^2 + c^2 - 2bc\cos A

the cosine rule

Symbols

$a$	side opposite angle A
$b,\ c$	sides forming angle A

Example

$b = 5$ , $c = 7$ and $A = 60^\circ$ : $a^2 = 25 + 49 - 70\cdot 0.5 = 39$ , so $a \approx 6.24$ .

Use the cosine rule first when all three sides are known to avoid ambiguity.

Practise trigonometry for free →

← The sine rule · The area rule →

Part of Mathematics 1T: Trigonometry.