Cosine Rule

## Cosine Rule

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. Make sure that the length on the left hand side of the formula (a) is opposite the angle stated on the right hand side of the formula (cos A).

## Example 1

What is the length of BC? Give your answer to 2 decimal places. Two sides and an included angle given, use the Cosine rule.

 COSINE rule: a^2 = b^2 + c^2 - 2bc cosA substitute a^2 = (12)^2 + (14)^2 - 2(12)(14) cos(76) a^2 = 144 + 196 - 336 xx 0.24192 a^2 = 258.71488 a = 16.0846 to 2dp a = 16.08

What is the size of the angle x? Give your answer to 1 decimal place. COSINE rule: a^2 = b^2 + c^2 - 2bc cosA substitute (10)^2 = (8)^2 + (9)^2 - 2(8)(9) cosA 100 = 155 - 144 cosA -55 = -144 cosA 0.3819 = cosA cos-1 both sides 67.546 = A to 1dp 67.6 = A