# Cosine Rule

In Trigonometry, the law of cosines is also known as Cosine Formula or Cosine Rule. These rules relate with the lengths of the sides of a triangle with any of its angle being a cosine angle.

### Laws of Cosine or Cosine Rules

As per the diagram, Cosine rules to find the length of the sides a, b & c of the triangle are –

$a^{2} = b^{2} + c^{2} -2bc \cos x$

$b^{2} = a^{2} + c^{2} -2ac \cos y$

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

To find the angles x, y & z, these formulae can be re-written as :

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

$\cos y = \frac{a^{2} + c^{2} -b^{2}}{2ac}$

$\cos z = \frac{a^{2} + b^{2} – c^{2}}{2ab}$

### Example -Find the length x in the following figure.

Solution: By applying the Cosine rule, we get:

$x^{2} = 22^{2} +28^{2} – 2.22.28 \cos 97$

$x^{2} = 1418.143$

$x^{2} = 1418.143$<