The cosine rule

Use the cosine rule to find a side from two sides and the included angle, or an angle from three sides, and see how it extends Pythagoras' theorem.

The rule

Cosine rule

$$c^2 = a^2 + b^2 - 2ab\cos C \qquad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$$

Pythagoras with a correction

C is at the origin and B is at (6, 0). Drag A. The cosine rule always gives the true length AB; Pythagoras only does when angle C is 90°.

Try this

Example 1: a side●

Find $c$ if $a = 5$, $b = 8$ and $C = 60^\circ$.

$c^2 = 25 + 64 - 2(5)(8)\cos 60^\circ$

$c^2 = 89 - 40 = 49$, so $c = 7$.

cos 60° = ½.

Example 2: an angle●●

Find the largest angle of the triangle with sides $7$, $8$ and $13$.

$\cos C = \dfrac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \dfrac{-56}{112} = -\dfrac12$

The largest angle is opposite the longest side.

$C = 120^\circ$

You know	Use
Two sides and the included angle	Cosine rule (for the third side)
Three sides	Cosine rule (for an angle)
A side and its opposite angle, plus one more	Sine rule

Flashcards

Cosine rule for a side

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

Cosine rule for an angle

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

Negative cos C

$C$ is obtuse.

Three sides known

Cosine rule.

Practice

Find $c$ if $a = 10$, $b = 12$ and $C = 110^\circ$, to 1 decimal place.
Find the largest angle of the triangle with sides $3$, $4$ and $5$.
Find the largest angle of the triangle with sides $5$, $6$ and $7$, to the nearest degree.
A triangle has sides $5$, $7$ and $10$. Which rule finds its angles?
$\cos C = -0.3$. Then $C$ is:
Find the third side of a triangle with sides $4$ and $6$ and included angle $35^\circ$, to 1 decimal place.
The cosine rule becomes Pythagoras' theorem when:
In a triangle with sides $a = b = c$, the cosine rule gives $\cos C =$