Cosine rule

Martin McBride

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:

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

This means that if we know the sides b and c, plus the enclosed angle A, we can find the other side, a.

Rearranging this gives:

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

This form of the equation means that if we know all three sides, we can find one of the angles.

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.