Cosine rule

The cosine rule can be used to find:

• the length of a side of a triangle when we know the lengths of the other two sides and the angle between them
• or the angle between two sides of a triangle when we know the lengths of all three sides.

The triangle

The lengths a, b and c represent the side lengths in this triangle.
The angles A, B and C represent the angles in the triangle.

            A
          /   \
     c  /       \  b
      /           \
    /               \
   B ---------------- C
            a

The cosine rule formula

Using the triangle above, the cosine rule tells us that:

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

And, equivalently:

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

Example

Find the length of side a

            75
          /    \
   8cm  /        \  5cm
      /            \
    /                \
   B ----------------- C
            a

• b = 8
• c = 5
• A = 75^\circ
• a^2 = b^2 + c^2 - 2bc \cos A
• a^2 = (8cm)^2 + (5cm)^2 - 2 \times 8cm \times 5cm \times \cos 75^\circ
• a^2 \approx 8.26cm