Sides over the sine of their opposite angles: For any triangle, each side divided by the sine of its opposite angle is the same: a/sinA = b/sinB = c/sinC. (Side a is opposite angle A.)
Interactive: tap Sine rule, Cosine rule or Area to see which sides and angles each formula uses on a general (non-right) triangle.
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Flip it to find an angle: To find an angle, use the rule upside down: sinA/a = sinB/b — it keeps the unknown sine on top.
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Need a matching pair: The sine rule works when you have a side with its opposite angle, plus one more piece. Set up two equal fractions and cross-multiply.
IB-style question — find a side
In triangle ABC, A = 40°, B = 75°, and side a = 10. Find side b.
Step by step
- Sine rule with the two pairs.
- Solve for b.
Final answer
b ≈ 15.0.
Sine rule — match each side with the angle OPPOSITE it: 10 is opposite 40°, and b is opposite 75°. So b/sin 75° = 10/sin 40° → b = 10 sin 75° ÷ sin 40° ≈ 15.0.
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IB-style question — find an angle
In triangle ABC, a = 8, A = 50°, b = 6. Find angle B.
Step by step
- Flip the rule.
- Solve.
Final answer
B ≈ 35.1°.
Sine rule for an angle: 8 is opposite 50°, and 6 is opposite the unknown B. So (sin B)/6 = (sin 50°)/8 → sin B = 6 sin 50° ÷ 8 → B ≈ 35.1°.
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Pythagoras with a correction term: The cosine rule generalises Pythagoras to any triangle: a² = b² + c² − 2bc·cosA, where A is the angle opposite side a. (When A = 90°, cosA = 0 and it becomes Pythagoras.)
Tap Cosine rule: it uses the two sides (b, c) either side of angle A and the side a opposite it — exactly the letters in a² = b² + c² − 2bc·cos A.
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The angle and side must match: In a² = b² + c² − 2bc·cosA, the side a on the left is opposite the angle A in the cos term.
SAS → side; SSS → angle: Use the cosine rule to find the third side from two sides and the included angle (SAS), or to find an angle from three sides (SSS) using the rearranged form.
IB-style question — find a side (SAS)
A triangle has b = 7, c = 9, and the angle A between them = 60°. Find a.
Step by step
- Cosine rule.
- Evaluate (cos 60° = ½).
Final answer
a ≈ 8.19.
Cosine rule (two sides + the angle BETWEEN them): sides 7 and 9 meet at 60°, and a is the side opposite that angle. a² = 7² + 9² − 2(7)(9)cos 60° → a ≈ 8.19.
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IB-style question — find an angle (SSS)
A triangle has sides a = 6, b = 5, c = 4. Find angle A.
Step by step
- Rearranged cosine rule.
- Inverse cosine.
Final answer
A ≈ 82.8°.
Cosine rule for an angle (all three sides known): A is opposite the side 6. cos A = (5² + 4² − 6²)/(2·5·4) = 0.125 → A ≈ 82.8°.
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Match the rule to the information: Right-angled? → SOH-CAH-TOA. Two sides + included angle (SAS) or three sides (SSS) → cosine rule. A side with its opposite angle → sine rule.
Cosine rule when…
- SAS — two sides + the angle between
- SSS — all three sides (to find an angle)
- no side–angle pair available
Sine rule when…
- you have a side opposite a known angle
- + one more side or angle
- often AAS or ASA setups
Tap each rule to see what it needs: Sine rule highlights a side with its OPPOSITE angle; Cosine rule highlights two sides + the angle between (or all three sides). Match the highlighted pieces to what the question gives you.
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No pair? Cosine first: If you can't see a side-with-its-opposite-angle, start with the cosine rule — it usually unlocks a pair you can then use with the sine rule.