The sine and cosine rules
Maths · GCSE · The School
Where right-angled trigonometry stops
SOH CAH TOA needs a right angle. Take it away and none of those ratios apply, which is why a triangle with angles of 50, 60 and 70 degrees defeats them entirely. Two rules replace them and work in any triangle. The labelling convention matters and is easy: a capital letter is an angle, the small letter of the same name is the side OPPOSITE that angle. So side a is opposite angle A. Getting that pairing wrong is the most common way these questions fail before any arithmetic happens.
The sine rule
a/sin A = b/sin B = c/sin C. Use it whenever you have a matching pair — a side and the angle opposite it — plus one more piece of information. For a missing side, keep the sides on top. For a missing angle, turn it upside down and keep the sines on top: sin A/a = sin B/b. One warning for the higher paper: when you use it to find an angle, a calculator returns the acute answer, and the obtuse angle 180 minus that answer may fit the triangle just as well. If the diagram or the question suggests an obtuse angle, check both.
The cosine rule
a² = b² + c² − 2bc cos A. Use it in the two cases the sine rule cannot start: three sides given, or two sides with the angle BETWEEN them. Rearranged for an angle it becomes cos A = (b² + c² − a²) / (2bc), which is the form to reach for when all three sides are known. Note what happens if A is 90 degrees: cos 90 is zero, the last term disappears, and the whole thing collapses to Pythagoras. The cosine rule is Pythagoras with a correction term for the angle not being square.
The decision is short. Three sides, or two sides and the angle between them, means cosine rule. Anything with a matching side-and-opposite-angle pair means sine rule. A triangle has sides 8 cm and 11 cm with an angle of 40 degrees between them; find the third side. Two sides and the included angle, so cosine rule: a² = 64 + 121 − 2 × 8 × 11 × cos 40 = 185 − 176 × 0.766 = 50.2, giving 7.09 cm. The area of that same triangle needs no height: area = ½ab sin C = ½ × 8 × 11 × sin 40 = 28.3 cm².