The Sine and Cosine Rules for Non-Right Triangles

1. 1 Show a triangle with no right angle, labelled with some sides and angles. 'Can we use SOH-CAH-TOA here?' (No.) 'We need new rules for any triangle.' Introduce the standard labelling: sides a, b, c opposite to angles A, B, C.
2. 2 a/sin A = b/sin B = c/sin C. 'Use the sine rule when you have a side and its opposite angle — plus one more piece of information.' Demonstrate: find side b given a = 8cm, A = 40°, B = 65°. Set up: 8/sin40° = b/sin65°. b = 8 sin65°/sin40° ≈ 11.3cm.
3. 3 Rearrange to find an angle: sin A/a = sin B/b. Given a = 7, b = 9, B = 50°: sin A = 7 sin50°/9 ≈ 0.596. A = sin⁻¹(0.596) ≈ 36.6°. Discuss the ambiguous case: sin A = 0.596 also gives A ≈ 143.4° — which is valid?
4. 4 a² = b² + c² − 2bc cos A. 'Use when you have two sides and the included angle (SAS) or all three sides (SSS).' Find a given b=7, c=5, A=60°: a² = 49+25−2(7)(5)cos60° = 74−35 = 39. a ≈ 6.24cm.
5. 5 Rearrange: cos A = (b²+c²−a²)/(2bc). Given a=8, b=6, c=5: cos A = (36+25−64)/60 = -3/60 = -0.05. A = cos⁻¹(-0.05) ≈ 92.9°. Note: negative cosine → obtuse angle.
6. 6 Area = ½ ab sin C. Find area of triangle with a=10, b=7, C=48°: Area = ½(10)(7)sin48° ≈ 26.0cm². Students apply to two more triangles.
7. 7 Give a series of unlabelled problems. Students identify: is it SAS? → cosine rule. Two angles and a side? → sine rule. All three sides? → cosine rule for angle. Practise rule selection before calculating.

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