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.

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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.

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Solution: By applying the Cosine rule, we get:

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

\(x^{2} = 1418.143\)

\(x^{2} = 1418.143\)<

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Practise This Question

Find value of x,y in the below given pair of equation.

2x + 3y = 13

5x - 4y = -2