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.
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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 | Missing 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°.
What a bearing means: A bearing is a direction measured clockwise from North, always written with three figures — 040°, not 40°.
To solve a bearings journey: sketch it, draw a North line at each turn, find the interior angle of the triangle, then use the cosine rule (for a distance) or the sine rule (for an angle, e.g. the bearing back).
IB-style question — bearings navigation
A yacht sails 7 km on a bearing of 040°, then 10 km on a bearing of 100°.
(a) Find the direct distance of the yacht from its starting point.
(b) Find the bearing it must steer to sail straight back to the start.
Step by step
- (a) Sketch the journey and mark the North line at the turn. The yacht arrives heading 040°, so it looks back along 040° + 180° = 220°, and leaves on 100°. The interior angle of the triangle is the difference.
- Apply the cosine rule to the two legs (7 and 10) with 120° between them.
- Square root.
- (b) Find the angle at C (between the return leg and the second leg) with the sine rule.
- At C the yacht faces 100°, so it looks back toward B along 100° + 180° = 280°. The start A lies 24.2° before that (see the diagram), giving the bearing of A from C.
Final answer
(a) ≈ 14.8 km. (b) ≈ 256°.
The journey from the question — 7 km on 040°, then 10 km on 100°. North lines are dashed at each turn; the purple dashed leg is the direct distance back to the start (part a).
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