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.)
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.
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.
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°.
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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.)
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.
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°.
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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
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.