Cosine Rule Revision Notes for Leaving Cert Mathematics

Cosine Rule

Two sides of the triangle and the included angle (i.e., the angle between the two sides).

The Cosine Rule for finding an unknown side is:

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

Where:

infoNote

Example:
Find the length of side $x$ in the triangle where $b = 5.2\text{ m}$ , $c = 4.5\text{ m}$ , and $\angle A = 58^\circ$ .

x^2 = 5.2^2 + 4.5^2 - 2 \times 5.2 \times 4.5 \times \cos 58^\circ

x^2 = 27.04 + 20.25 - 23.4 \times 0.5299

x^2 = 22.48977\ \text{m}^2

x = \sqrt{22.48977} \approx 4.74\ \text{m}\ (2\text{ dp})

So, the length of side $x$ is approximately $4.74\ \text{m}$ .

The Cosine Rule for finding an unknown angle is:

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

Where:

infoNote

Example:
Find the size of angle $x$ in the triangle where $a = 9\text{ m}$ , $b = 11\text{ m}$ , and $c = 12\text{ m}$ .

\cos x = \frac{11^2 + 12^2 - 9^2}{2 \times 11 \times 12}

\cos x = \frac{121 + 144 - 81}{264} = \frac{184}{264} \approx 0.69697

x = \cos^{-1}(0.69697) \approx 72.97^\circ\ (2\text{ dp})

So, the angle is approximately $72.97^\circ$ .