Sine, Cosine Rule & Area of Triangles

• Sine Rule connects a side and its opposite angle — use it when you have an opposite pair (side + angle across from it).
• Cosine Rule works with all three sides, or two sides + the angle between them — no opposite pair needed.
• Area formula (½ab sin C) needs two sides and the included angle between them.
• The Ambiguous Case of the Sine Rule means sometimes there are two valid triangles from the same information — your calculator only gives you one of them.
• Harder questions often need more than one rule chained together, or the fact that angles in a triangle sum to 180°.
• A simple flowchart (right-angled? opposite pair? area?) tells you exactly which rule to pick every time.

Picture any triangle — not necessarily right-angled — with corners labelled A, B, and C. The convention (and it matters!) is: capital letters for angles, and the lowercase letter of the side directly opposite that angle. So side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

The Sine Rule says that if you take any side and divide it by the sine of its own opposite angle, you get the same number no matter which side/angle pair you pick. That's genuinely elegant — it means these three ratios are all secretly equal.

Use the Sine Rule whenever the question gives you an opposite pair — meaning a side and the angle directly across from it (like a and A), plus one more piece of information (another side or another angle) that lets you build a full ratio.

Finding a missing length? Use the un-flipped version (sides on top) — you only ever need to set two of the three ratios equal, not all three at once.

Finding a missing angle? Flip the rule first so the angle is on top, substitute in, then finish with sin⁻¹(...) on your calculator.