What's the cosine rule?

The Rule is given to you on the formula sheet in the GCSE exam paper:

a2 = b2 + c2 – 2bc(Cos A)

The formula may look complicated but all you need to do is plug in what you know, to get what you want! 

Where a, b and c are the sides of triangles, and A is the angle opposite side a.

What do we use it for?

The Cosine Rule is a more advanced formula that be used to find missing sides and angles in any triangle, whether it has a right angle or not.

When can we use it?

If you know 2 sides and one angle, and want to find the missing side: as long as you label the known angle A, and the side opposite a, you can use the formula and rearrange to get the missing side.

If you know all 3 sides and want to find a missing angle: as long as you label the missing angle A, and the opposite side a, you can rearrange the formula using the inverse cos function to find A:

A = - Cos-1[(a2 - b2 - c2) / 2bc]

Remember! Cos-1 does not mean Cos to the power -1! It means inverse Cos


Note! The following is interesting, but not needed for the exam: If you ignore everything after the c2, this should already look very familiar to you. This may or may not be taught by your teacher, but the Cosine Rule is also an advanced form of Pythagoras’ Theorem (a2 = b2 + c2, where a is the hypotenuse). Pythagoras’ theorem can only be used in right angled triangles – if you try to use the Cosine Rule in a right angled triangle, then angle A would be 90 degrees. Type ‘Cos 90’ into a calculator and you will discover that Cos 90 = 0. This means in a right angled triangle, everything after the c2 disappears, since anything x 0 just becomes 0, leaving you with the familiar Pythagoras’ theorem.

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