Cosine Rule

Cosine Rule

GCSE(H),

The cosine rule can be used for any triangle. The letters relating to the side are opposite the corresponding letters for the angles, i.e. a and A; b and B and c and C.

The rule is:

`a^2 = b^2 + c^2 - 2bc cosA`

Use the cosine rule when all three sides are known, or two sides and an included angle are known.

Examples

1. What is the length of BC? Give your answer to 2 decimal places.

Answer: 16.08 cm

Using the cosine rule:

`a^2 = b^2 + c^2 - 2bc cosA`

`a^2 = 12^2 + 14^2 - 2 xx 12 xx 14 xx cos 76`

`a^2 = 144 + 196 - 336 xx 0.24192`

`a^2 = 340 - 81.28512`

`a^2 = 258.71488`

`a = 16.0846`, or 16.08 (2dp)

2. What is the size of the angle x? Give your answer to 1 decimal place.

Answer: 67.5ยบ

Using the cosine rule:

`a^2 = b^2 + c^2 - 2bc cosA`

`10^2 = 8^2 + 9^2 - 2 xx 8 xx 9 xx cos A`

`100 = 64 + 91 - 144 xx cos A`

`100 = 155 - 144 xx cos A`

`-55 = -144 xx cos A`

`cos A = 0.3819`

`A = cos^-1 0.3819`

`A = 67.546`, or 67.5 (1dp)