Cosine Rule

Statement

The Cosine Rule relates the lengths of the sides of any triangle to the cosine of one of its angles. It generalises Pythagoras’ theorem to non-right-angled triangles:

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

Here \(a\) is the side opposite angle \(A\), and \(b\), \(c\) are the other two sides.

Why it’s true

• Drop a perpendicular from the vertex opposite \(a\) to create two right-angled triangles.
• Apply Pythagoras’ theorem and basic trigonometry to express the base segments in terms of \(\cos A\).
• Simplify to get \(a^{2} = b^{2} + c^{2} - 2bc\cos A\).

Recipe (how to use it)

1. Label the triangle so that side \(a\) is opposite angle \(A\).
2. Substitute the known sides and the included angle into the formula.
3. Rearrange if needed:
   • To find an angle: \(\cos A = \dfrac{b^{2} + c^{2} - a^{2}}{2bc}\).
   • To find a side: use the formula as written and take a square root.
4. Check that your calculator is in the correct mode (degrees or radians).

Spotting it

Use the Cosine Rule when you know:

• Two sides and the included angle (SAS) and need the third side, or
• All three sides (SSS) and need an angle.

Common pairings

• Pythagoras’ theorem (\(A=90^\circ\) gives \(a^2=b^2+c^2\)).
• Sine Rule for other triangle problems.

Mini examples

1. Find \(a\) when \(b=7\), \(c=9\), \(A=60^\circ\): \(a^2 = 7^2 + 9^2 - 2\cdot7\cdot9\cos60^\circ = 49 + 81 - 126\cdot0.5 = 130 - 63 = 67\), so \(a \approx 8.19\).
2. Find \(A\) when \(a=10\), \(b=8\), \(c=7\): \(\cos A = \dfrac{8^2 + 7^2 - 10^2}{2\cdot8\cdot7} = \dfrac{64 + 49 - 100}{112} = \dfrac{13}{112}\), so \(A \approx 83.3^\circ\).

Pitfalls

• Mixing the letters: side \(a\) must be opposite angle \(A\).
• Forgetting to take the square root after finding \(a^2\).
• Calculator in wrong mode (degrees vs radians).

Exam strategy

• Write the formula before substituting numbers.
• Round only at the end for accuracy.
• Check that answers are sensible (e.g. side lengths positive, angles between 0° and 180°).

Summary

The Cosine Rule allows you to find missing sides or angles in any triangle when you know two sides and the included angle or all three sides. It extends Pythagoras’ theorem to all triangles.