Sine rule and cosine rule
Edexcel 1MA1 Higher papers regularly test the sine and cosine rules in multi-step problems, often combined with the area formula ½ab sin C.
The sine rule
$$ rac{a}{sin A} = rac{b}{sin B} = rac{c}{sin C}$$
(or the inverted form for finding angles: sin A/a = sin B/b = sin C/c)
Use when you know: an angle–opposite-side pair, plus one other side or angle (ASA or SSA).
Finding a side: a = b sin A / sin B. Finding an angle: sin A = a sin B / b.
Ambiguous case (SSA): when given two sides and a non-included angle, there may be two possible triangles. Edexcel may test this at Higher — always check if sin A > 1 (no solution) or if there are two valid solutions.
The cosine rule
Finding a side (SAS): $$a^2 = b^2 + c^2 - 2bc cos A$$
Finding an angle (SSS): $$cos A = rac{b^2 + c^2 - a^2}{2bc}$$
Use when you know: two sides and the included angle (SAS), or all three sides (SSS).
Area of a triangle using ½ab sin C
$$ ext{Area} = rac{1}{2}absin C$$
where C is the angle between sides a and b. Works for any triangle.
Choosing the right formula
| Known information | Formula to use |
|---|---|
| Right angle present | Pythagoras / SOHCAHTOA |
| Two sides + included angle (SAS) | Cosine rule (for side) |
| Three sides (SSS) | Cosine rule (for angle) |
| Angle-side pair + one more side or angle | Sine rule |
| Two sides + included angle (area) | ½ab sin C |
⚠Common mistakes
- Using sine rule when cosine rule is needed (SAS → cosine rule, not sine rule).
- Forgetting to take the square root in the cosine rule when finding the side.
- Using a² = b² + c² + 2bc cos A (forgetting the minus sign) — it is minus.
- Calculator mode — radians will give completely wrong answers.
- Not checking for the ambiguous case when using the sine rule to find an angle.
AI-generated · claude-opus-4-7 · v3-edexcel-maths