The Sine and Cosine Rules

The method, step by step

1. 1

   Check whether the triangle has a right angle

   right angle → use SOHCAHTOA, not these rules

   SOHCAHTOA is faster and simpler where it applies. The sine and cosine rules exist for triangles where it does not, so establishing which case you are in is the first decision.

2. 2

   Look for a matching side and opposite angle

   a/sin A = b/sin B = c/sin C

   The sine rule needs a complete pair — a side and the angle directly opposite it. If you have one and can identify a second known value from another pair, the sine rule solves it.

3. 3

   Use the cosine rule when no pair is complete

   a² = b² + c² − 2bc·cos A

   Two sides with the angle between them, or all three sides, means no complete pair exists and the cosine rule is the only route. Note that when A is 90°, cos A is 0 and this reduces to Pythagoras.

4. 4

   Rearrange the cosine rule to find an angle

   cos A = (b² + c² − a²) / 2bc

   Given three sides, rearranging gives the angle. The side a must be the one opposite the angle A you are finding, and getting that pairing wrong is the most common error with this rule.

5. 5

   Use the area formula when you have two sides and the included angle

   Area = ½ab·sin C

   This works for any triangle and needs no perpendicular height. The angle C must be between the two sides a and b, which is the condition students most often overlook.

How to decide which rule

The decision rests entirely on what you have been given. If you can see a side and the angle opposite it, the sine rule applies. If you cannot — because the angle you know sits between two known sides, or because you know all three sides — it must be the cosine rule.

Marking the given values on the diagram before choosing anything makes this decision almost automatic. Students who try to choose the rule before annotating the triangle frequently pick the one they remember better rather than the one that fits.

• Right angle present → SOHCAHTOA
• Side and opposite angle known → sine rule
• Two sides and included angle → cosine rule
• All three sides → cosine rule for an angle
• Area = ½ab·sin C, with C between a and b

The ambiguous case

When the sine rule is used to find an angle, two answers are possible, because sine gives the same value for an angle and for 180° minus that angle. A calculator returns only the acute one.

Whether the obtuse answer is valid depends on the triangle — the three angles must still total 180°. Questions sometimes require both solutions, and students who accept the calculator's answer without checking lose those marks.

The cosine rule contains Pythagoras

Setting A to 90° makes cos A zero, so a² = b² + c² − 0, which is Pythagoras. The cosine rule is therefore a generalisation rather than a separate result, with Pythagoras as the special case where the angle happens to be right.

Pointing this out reduces what has to be remembered and gives students a way to check they have written the formula correctly — if setting the angle to 90° does not produce Pythagoras, something has been misremembered.

How we teach these rules

We teach the decision before the formulae. Students who know both rules perfectly and choose the wrong one score nothing, and choosing correctly is a two-second check against what the diagram provides.

We derive the Pythagoras connection in front of the student. It takes a minute, it makes the cosine rule less arbitrary, and it gives them a self-check that survives exam pressure better than memory alone.