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

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}

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

Question	Answer
What does the cosine rule allow you to find?	It allows you to find the length of a side when you know the other two sides and the included angle, or an angle when you know all three sides.
What is the formula for the cosine rule for side a?	a^2 = b^2 + c^2 - 2bc \cos A
What is the equivalent formula for \cos A derived from the cosine rule?	\cos A = \frac{b^2 + c^2 - a^2}{2bc}
In the triangle notation, what do a, b, and c represent?	They represent the lengths of the sides.
In the triangle notation, what do A, B, and C represent?	They represent the angles opposite the sides a, b, and c respectively.
In the example, which values are given for side lengths and the included angle?	b = 8\text{cm}, c = 5\text{cm}, and A = 75^\circ
In the example, what is the calculated approximate value of side a^2?	a^2 \approx 8.26\text{cm}^2 (reading as 8.26\text{cm}^2)