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 , and represent the side lengths in this triangle.
The angles , and 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:

And, equivalently:

Example

Find the length of side

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