Sine rule (find a side) $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$use when you have a side and its OPPOSITE angle, plus one more of either Sine rule (find an angle) $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$flip it up the other way when the unknown is an angle Cosine rule (find a side) $a^2=b^2+c^2-2bc\cos A$use for two sides and the INCLUDED angle (SAS) Cosine rule (find an angle) $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$rearranged, for three sides (SSS) Area of a triangle $\text{Area}=\tfrac12\,ab\sin C$two sides and the INCLUDED angle $C$ Sine rule · $A=40^\circ,\ B=75^\circ,\ a=8$
$\dfrac{b}{\sin75^\circ}=\dfrac{8}{\sin40^\circ}$
$b=\dfrac{8\sin75^\circ}{\sin40^\circ}=12.0$ cm
Cosine rule · $b=7,\ c=9,\ A=50^\circ$
$a^2=7^2+9^2-2(7)(9)\cos50^\circ$
$a^2=49.0\ldots$, so $a=7.0$ cm
Area · sides $10$ and $8$, angle $30^\circ$
$\text{Area}=\tfrac12(10)(8)\sin30^\circ$
$=20.0$ cm$^2$
Sine rule: links each side to the sine of its opposite angle; used with an opposite side-angle pair.
Cosine rule: links all three sides and one angle; used for SAS (a side) or SSS (an angle).
Included angle: the angle BETWEEN two known sides – needed for the cosine rule (SAS) and the area formula.
Opposite side: the side across the triangle from an angle; side $a$ is opposite angle $A$.
Ambiguous case: the sine rule can give two possible angles ($\theta$ and $180^\circ-\theta$).
✗ Using the sine rule for SAS (two sides + the angle between)
✓ no opposite pair, so use the COSINE rule.
✗ $a^2=b^2+c^2+2bc\cos A$
✓ it is MINUS: $a^2=b^2+c^2-2bc\cos A$.
✗ Calculator in radians
✓ set it to DEGREES for GCSE angles.
✗ Area $=\tfrac12 ab\sin C$ with a non-included angle
✓ $C$ must be the angle BETWEEN sides $a$ and $b$.
• Side $a$ is always OPPOSITE angle $A$ (same letter).
• Choose the cosine rule when you have no complete opposite side-angle pair.
• The angles of the triangle still add to $180^\circ$ – use it to find a third angle.
• Label the triangle: put $a$ opposite $A$, etc., before choosing a rule.
• Opposite pair given? Sine rule. Otherwise? Cosine rule.
• Keep full accuracy on your calculator; round only the final answer.
mathedup.co.uk · sheet B7IN