• 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.

Here's something genuinely surprising: sometimes, the exact same information (two sides and an angle that isn't between them) can be used to build two completely different triangles. Imagine you're given side a, side b, and base angle θ — but no angle near side b. Side b could "swing" to two different positions and still satisfy all the given measurements, like a compass arm that can land in two different spots while staying the same length.

The catch: your calculator's sin⁻¹ button will only ever give you the acute angle answer. It physically cannot output an obtuse angle from an inverse sine calculation. So if the real diagram calls for the obtuse version, you have to spot it yourself and calculate it manually.

The Sine Rule needs an "opposite pair" to work. But what if you don't have one? What if you're given, say, two sides and the angle squeezed between them — with no angle-side pair facing each other at all? That's exactly when the Cosine Rule steps in. Think of it as a generalised version of Pythagoras' theorem that also accounts for the angle between the two known sides (Pythagoras is actually just a special case of it, when that angle is 90°!).

If you're given all three sides and need to find any angle, you rearrange the formula. Watch how it unfolds — you add 2bc cos A to both sides, then divide to isolate cos A:

You already know Area = ½ × base × height for a triangle. The problem: in a non-right-angled triangle, you often don't have the height — nobody's drawn that perpendicular line for you. But if you know two sides and the angle trapped between them, you can find the area directly, no height-hunting required.

Knowing the formulas is only half the battle — IGCSE exam questions are designed to test whether you can look at what you're given and what you need to find, and correctly pick the tool. This decision table is worth memorising cold:

• Forgetting to square root after using the Cosine Rule to find a side. You solve for a², not a — don't submit the squared value as your final answer!
• Using the wrong angle in the Area formula. The angle you use MUST be physically between the two sides you're multiplying — not just any angle given in the question.
• Ignoring the ambiguous case. If a Sine Rule question gives two sides and a non-included angle, always sanity-check your diagram — the "real" angle might be obtuse even though your calculator gave you an acute one.
• Mixing units. A classic trap is giving one side in cm and another in m — always convert to the same unit before you substitute anything into a formula.
• Getting a calculator "Math Error" on cos⁻¹ usually means you've rearranged the Cosine Rule incorrectly — double check your algebra (a common slip is a sign error when moving terms across the equals sign).
• Look at the mark allocation. If a question is worth 4-5+ marks, it's very likely you'll need more than one trig rule (or the angle sum of a triangle) to get all the way to the answer.
• Relabel your triangle every time. Don't try to memorise the formula only in terms of A, B, C — get comfortable renaming any triangle's vertices to match whichever formula you're about to use.
• Both the Sine Rule (sides version), Cosine Rule, and Area formula are given on your formula sheet — you don't need to memorise those three from scratch. But the flipped Sine Rule (for angles) and the rearranged Cosine Rule (for angles) are NOT given, so practise deriving/recalling those.
• Draw a quick sketch even if the question doesn't require one — labelling sides and angles visually makes it far easier to spot which rule applies and avoid mixing up which angle is "included."