Sine Rule, Cosine Rule and Area of a Triangle
These three formulae handle triangles that have no right angle. The mathematics is short; the difficulty is almost entirely in choosing the right rule, which is where most marks in this topic are won and lost.
1. Labelling — the convention everything depends on
Capital letters (A, B, C) are the angles. Lower-case letters (a, b, c) are the sides. Side a is opposite angle A, side b is opposite angle B, side c is opposite angle C.
This pairing of a side with the angle across from it is the whole basis of the sine rule. Label the triangle before you write anything else.
“Opposite” no longer means what it did in SOHCAHTOA. In a right-angled triangle, opposite/adjacent were defined relative to your chosen angle. Here, each side is simply paired with the angle facing it. Carrying the old meaning across is a common source of confusion.
2. Choosing the rule
This single table decides most of the marks:
| What you are given | Use |
|---|---|
| A matching pair (a side and its opposite angle) plus one more piece | Sine rule |
| Two sides and the angle between them (SAS) | Cosine rule |
| All three sides (SSS), and you want an angle | Cosine rule |
| Two sides and the angle between them, and you want the area | Area = ½ab sin C |
| There is a right angle | SOHCAHTOA / Pythagoras — simpler |
The sine rule needs a complete pair. You must know a side and the angle opposite it. Without that pair, the sine rule cannot start — trying to use it anyway was one of the most frequent errors in lessons.
The cosine rule needs the included angle — the angle between the two known sides. If the angle you have is not between them, it isn’t a cosine rule question.
Check for a right angle first. Several students applied the sine or cosine rule to a right-angled triangle, or Pythagoras to a triangle with no right angle. Look at the diagram before choosing.
3. The sine rule
To find a side:
a / sin A = b / sin B = c / sin C
To find an angle (flip it, to keep the unknown on top):
sin A / a = sin B / b = sin C / c
Example — finding a side. In triangle ABC, A = 40°, B = 75°, b = 10 cm. Find a.
- a / sin 40° = 10 / sin 75°
- a = 10 × sin 40° / sin 75°
- a = 6.65 cm (3 s.f.)
Example — finding an angle. a = 8 cm, A = 50°, b = 6 cm. Find B.
- sin B / 6 = sin 50° / 8
- sin B = 6 × sin 50° / 8 = 0.5745
- B = sin⁻¹(0.5745) = 35.1° (3 s.f.)
You only ever use two of the three fractions at once. Pick the pair containing your unknown and the pair you know completely.
The obtuse angle case
When you find an angle with the sine rule, there can be two possible answers: θ and 180° − θ.
This is because sin 35° and sin 145° are equal. If the question or diagram indicates the angle is obtuse, take 180° − θ.
Your calculator only ever gives you the acute answer. If the triangle clearly contains an obtuse angle, you must subtract from 180° yourself — the calculator will not warn you. This was a documented error, and it silently produces a wrong triangle.
4. The cosine rule
To find a side:
a² = b² + c² − 2bc cos A
To find an angle:
cos A = (b² + c² − a²) / (2bc)
Example — finding a side. b = 7 cm, c = 9 cm, A = 60°. Find a.
- a² = 7² + 9² − 2(7)(9) cos 60°
- a² = 49 + 81 − 126 × 0.5
- a² = 130 − 63 = 67
- a = √67 = 8.19 cm (3 s.f.)
Remember the square root. Forgetting it — leaving the answer as 67 — is the most common single error with this formula.
Example — finding an angle. a = 5, b = 6, c = 7. Find A.
- cos A = (6² + 7² − 5²)/(2 × 6 × 7) = (36 + 49 − 25)/84 = 60/84 = 0.7143
- A = cos⁻¹(0.7143) = 44.4° (3 s.f.)
The formula is a subtraction, not an addition. Writing b² + c² + 2bc cos A appeared repeatedly and gives an answer that is too large but looks reasonable.
You cannot simplify b² + c² − 2bc cos A before substituting. The term with cos A is not like the others; combine only after you have a number for cos A.
A negative cosine means an obtuse angle — that is correct and expected, not an error. cos⁻¹ of a negative value gives an angle between 90° and 180°.
The cosine rule is Pythagoras with a correction term. When A = 90°, cos 90° = 0 and the formula collapses to a² = b² + c². Seeing that makes it much easier to remember — and shows why Pythagoras fails in a non-right triangle.
5. Area of a triangle
Area = ½ ab sin C
where a and b are two sides and C is the angle between them.
Example: a = 8 cm, b = 11 cm, included angle C = 35°.
- Area = ½ × 8 × 11 × sin 35° = 25.2 cm² (3 s.f.)
The angle must be between the two sides. Using a different angle is the classic error here.
Use ½ × base × height only when you have a perpendicular height. In a non-right triangle you usually don’t, which is exactly why this formula exists.
Area units are squared — cm², m². Dropping the square costs the final mark.
6. Method and accuracy
- Draw the triangle if one isn’t given, and label sides and angles properly
- Decide which rule the given information allows
- Write the formula before substituting
- Substitute, then solve
- Check the answer is sensible — the longest side must face the largest angle
Draw a diagram even when the question is in words. Tutors made this point constantly for bearings and navigation problems: the diagram is where the marks start.
Write the formula down before substituting. It earns method marks even if the arithmetic later goes wrong.
Keep full accuracy through the middle of a question. Round only the final answer, to 3 significant figures unless told otherwise. Rounding a mid-step value and reusing it is a reliable way to lose accuracy marks.
Check whether your formula list gives these rules. In lessons, tutors said the sine rule is provided in the exam — confirm against the formula sheet for your series so you know what you must memorise.
Calculator in degree mode, and use brackets round the whole numerator and denominator.
7. Mistakes that cost marks
Using the sine rule without a complete side–angle pair.
Using the cosine rule when the angle is not between the two sides.
Applying either rule to a right-angled triangle when SOHCAHTOA was simpler — or Pythagoras to a triangle with no right angle.
Forgetting the square root in the cosine rule.
Adding instead of subtracting 2bc cos A.
Combining terms before substituting for cos A.
Missing the obtuse case when finding an angle with the sine rule.
Using the wrong angle in ½ab sin C.
Mismatching sides and angles — side a must be opposite angle A.
Rounding too early.
Leaving off units, or forgetting that area is squared.
Frequently asked questions
When do I use the sine rule and when the cosine rule? Sine rule when you have a side with its opposite angle. Cosine rule when you have two sides and the angle between them, or all three sides.
What is the sine rule? a / sin A = b / sin B = c / sin C — flip it to sin A / a = … when finding an angle.
What is the cosine rule? a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²)/(2bc) for an angle.
What is the area formula for a non-right triangle? Area = ½ ab sin C, where C is the angle between sides a and b.
Why do I sometimes get two answers for an angle? Because sin θ = sin(180° − θ). If the angle is obtuse, use 180° − θ; your calculator only returns the acute value.
What if cos A comes out negative? The angle is obtuse. That is a valid result.
Can I use Pythagoras in these triangles? Only if there is a right angle. Otherwise use the cosine rule.
How does the cosine rule relate to Pythagoras? It is Pythagoras plus the term −2bc cos A. When A = 90°, that term is zero.
Quick revision checklist
- I label sides with lower case, angles with capitals, a opposite A
- I check for a right angle before choosing a rule
- I know the sine rule needs a complete side–angle pair
- I know the cosine rule needs the included angle, or all three sides
- I can use the sine rule for a side and (flipped) for an angle
- I check whether an angle should be obtuse (180° − θ)
- I can use the cosine rule both ways round
- I remember the square root
- I know −2bc cos A is subtracted, and can’t be simplified early
- I know a negative cosine means an obtuse angle
- I can use Area = ½ab sin C with the included angle
- I draw and label a diagram every time
- I write the formula before substituting
- I keep full accuracy and round only at the end
- I give units, squared for areas
These notes cover the sine rule, cosine rule and area of a triangle in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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