When to use the sine rule: Use the sine rule when you have an angle–opposite-side pair plus one more unknown. It works for any triangle, right-angled or not.
The lower-case letter is the side length; the upper-case letter is the angle opposite that side. Side a is opposite angle A, side b opposite B, and side c opposite C.
Worked example — find unknown side
In triangle ABC, angle A = 40°, angle B = 65°, and side a = 12 cm. Find side b.
Step by step
- Write the sine rule pairing the known and unknown.
- Substitute.
- Solve for b.
Final answer
b ≈ 16.9 cm.
When to use the cosine rule: Use the cosine rule when you have two sides and the included angle (to find the third side), or all three sides (to find any angle).
| Situation | Formula |
|---|---|
| Find side a (given b, c, A) | a² = b² + c² − 2bc cos A |
| Find angle A (given a, b, c) | cos A = (b² + c² − a²) / (2bc) |
Worked example — find unknown side
In triangle ABC, b = 7 cm, c = 9 cm, and angle A = 58°. Find side a.
Step by step
- Substitute into the cosine rule.
- Compute.
- Square root.
Final answer
a ≈ 7.95 cm.
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Area formula for any triangle: When you know two sides and the included angle, you can find the area without needing the perpendicular height.
Worked example — area
A triangle has sides a = 8 cm, b = 5 cm, and included angle C = 70°. Find its area.
Step by step
- Apply the formula.
- Calculate.
Final answer
Area ≈ 18.8 cm².
C is the angle between a and b: The angle C must be the included angle — the one between sides a and b. Labelling matters here.
| What you know | What you want | Use |
|---|---|---|
| Angle + opposite side pair, another side | Mi ing angle or side | Sine rule |
| Two sides + included angle | Third side | Cosine rule |
| All three sides | An angle | Cosine rule (rearranged) |
| Two sides + included angle | Area | ½ab sinC |
Worked example — find angle using cosine rule
A triangle has sides 5, 7, and 8. Find the largest angle.
Step by step
- The largest angle is opposite the longest side (8). Call it C.
- Find C.
Final answer
Largest angle ≈ 81.8°.