The cosine rule is a2=b2+c2-2bccos A, where A is the angle between b and c, opposite a. If A=90, cos A=0 and it becomes Pythagoras’ theorem. **Known information** | **Useful first method** Right angle and two sides | Pythagoras for another side; trigonometric ratio for an angle. Two angles and one side | Angle sum, then sine rule. Two sides and included angle | Cosine rule for the third side; sine-area formula for area. Three sides | Cosine rule for an angle. Two sides and non-included angle | Sine rule, with both candidate angles checked. Example Two sides are 5 cm and 7 cm with included angle 60. The opposite side is sqrt(52+72-2(5)(7)cos60)=sqrt(39) cm 6.24 cm. For sides 5 cm, 7 cm and 8 cm, the angle opposite 8 cm satisfies cos A= 52+72-822(5)(7)= 17, so A 81.8. A negative cosine would give an obtuse angle.

The cosine rule is \(a^2=b^2+c^2-2bc\cos A\), where \(A\) is the angle between \(b\) and \(c\), opposite \(a\). If \(A=90^\circ\), \(\cos A=0\) and it becomes Pythagoras’ theorem.

Two sides are 5 cm and 7 cm with included angle \(60^\circ\). The opposite side is \(\sqrt{5^2+7^2-2(5)(7)\cos60^\circ}=\sqrt{39}\text{ cm}\approx6.24\text{ cm}\).

For sides 5 cm, 7 cm and 8 cm, the angle opposite 8 cm satisfies \(\cos A=\dfrac{5^2+7^2-8^2}{2(5)(7)}=\dfrac17\), so \(A\approx81.8^\circ\). A negative cosine would give an obtuse angle.

The cosine rule and choosing a method

The cosine rule and choosing a method

Known information	Useful first method
Right angle and two sides	Pythagoras for another side; trigonometric ratio for an angle.
Two angles and one side	Angle sum, then sine rule.
Two sides and included angle	Cosine rule for the third side; sine-area formula for area.
Three sides	Cosine rule for an angle.
Two sides and non-included angle	Sine rule, with both candidate angles checked.