# Cosine rule

Categories: gcse trigonometry

The cosine rule is a trigonometry formula that relates the sides and angles of a triangle. It can be used to *solve a triangle* if we know either:

- Two sides of the triangle, and the angle enclosed between those sides.
- Three sides but none of the angles.

For other cases, you will need to use the sine rule.

The rule applies to any triangle, not just right-angled triangles.

## Labelling the triangle

It is important to label the triangle correctly, otherwise the rule won't work! We name the angles **A**, **B** and **C**, and we name the sides *a*, *b* and *c*:

The important thing to remember is that each angle is opposite the side of the same name:

- Angle
**A**is opposite side*a*. - Angle
**B**is opposite side*b*. - Angle
**C**is opposite side*c*.

## The cosine rule

The cosine rule tells us that:

## Finding a side using the cosine rule

In this example, we know the sides *b* and *c*, plus the enclosed angle **A**, and we wish to find the other side, *a*.

The cosine rule lets us find *a* squared, so to find *a* we need to take the square root of both sides. Here is the formula:

## Finding an angle using the cosine rule

In this example we know the sides *a*, *b* and *c*, so we can find the angle, **A**:

The cosine rule can be rearranged to find the cosine of **A**. Here is the formula:

In both cases, we will know the value of one angle and all three sides. We can then use the sine rule to find the other angles.

## See also

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