The law of cosines

Warm-up

Three quick questions from earlier lessons.

Discover

If the angle between sides 3 and 4 is 90°, the third side is 5. If the angle is bigger than 90°, the third side is…

Longer than 5
Still 5

√(5² + 8²) ≈ 9.43. Set C above 90°. How long is c?

Longer than 9.43
Still 9.43

Key idea

The law of cosines corrects the Pythagorean theorem.

In any triangle,

c^2 = a^2 + b^2 - 2ab \cos C

, where

C

is the angle opposite

c

. Use it with two sides and the angle between them (to find the third side) or three sides (to find an angle:

\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}

). For

C = 90°

it is the Pythagorean theorem; for an obtuse

C

,

\cos C < 0

and

c

gets longer.

Worked example

a = 5, b = 8, C = 60°.

c^2 = 25 + 64 - 2 \cdot 5 \cdot 8 \cdot \tfrac12

Formula.

c^2 = 49

,

c = 7

Result.

Your turn, guided

Your turn: a = 3, b = 5, C = 120°. What is c²?

49

9 + 25 + 15.

Practice ★

In triangle ABC, a = 12, b = 8 and the angle between them is C = 60°. Use the law of cosines to find c².
In triangle ABC, a = 3, b = 3 and the angle between them is C = 60°. Use the law of cosines to find c².

In triangle ABC, a = 12, b = 5 and C = 120°. Use the law of cosines to find c².

Watch out!

Practice ★★

A triangle has sides a = 5, b = 6 and c = 8. Find cos C, the cosine of the angle opposite c.
A triangle has sides a = 4, b = 8 and c = 6. Find cos C, the cosine of the angle opposite c.

Boss problem. Two hikers leave camp along straight paths that make a 60° angle. One walks 6 km, the other 8 km.

Exit ticket

Three questions to check you have got it.

In triangle ABC, a = 12, b = 8 and the angle between them is C = 60°. Use the law of cosines to find c².
In triangle ABC, a = 3, b = 3 and the angle between them is C = 60°. Use the law of cosines to find c².
In triangle ABC, a = 8, b = 7 and the angle between them is C = 60°. Use the law of cosines to find c².

Summary card

Law of cosines

c² = a² + b² − 2ab cos C.

Watch out: Pythagoras alone works only for a right angle.

About this lesson