Trigonometry
Sine, cosine and area rules, 2D problems
Sine Rule, Cosine Rule and Area Rule
For any triangle (not just right-angled):
• Sine rule: a/sin A = b/sin B = c/sin C
• Cosine rule: a² = b² + c² − 2bc·cos A
• Area rule: Area = ½ab·sin C
Example
Using the Sine Rule
In △ABC: A = 40°, B = 60°, a = 10 cm. Find b.
a/sin A = b/sin B
10/sin 40° = b/sin 60°
b = 10 × sin 60° / sin 40°
b = 10 × 0.866 / 0.643
b ≈ 13.47 cm
Example
Using the Cosine Rule
In △PQR: p = 7, q = 9, R = 120°. Find r.
r² = 7² + 9² − 2(7)(9)cos 120°
r² = 49 + 81 − 126(−0.5)
r² = 130 + 63 = 193
r = √193 ≈ 13.89
Note
Remember
Use sine rule when you have angle-side pairs. Use cosine rule when you have SAS or SSS. The area rule gives the area when you know two sides and the included angle. These work for ALL triangles.
Key Vocabulary
Sine rulea/sin A = b/sin B = c/sin C
Cosine rulea² = b² + c² − 2bc·cos A
Area ruleArea = ½ab·sin C
Included angleThe angle between two known sides
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Sine rule
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