Solutions — The Cosine Rule and Area Formula

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1. Show Solution
   c² = a² + b² − 2ab cos C
   c² = 6² + 8² − 2(6)(8) cos 45°
   c² = 36 + 64 − 96 × 0.70711
   c² = 100 − 67.88 = 32.12
   c = √32.12 ≈ 5.67 cm
2. Show Solution
   cos R = (p² + q² − r²) / (2pq)
   cos R = (16 + 49 − 81) / (2 × 4 × 7)
   cos R = −16 / 56 = −0.28571
   R = cos⁻¹(−0.28571) ≈ 106.6°
3. Show Solution
   s = (5 + 8 + 11)/2 = 12
   Area = √(s(s−a)(s−b)(s−c))
   = √(12 × 7 × 4 × 1)
   = √336 ≈ 18.3 m²
4. Show Solution
   a² = b² + c² − 2bc cos A
   a² = 100 + 169 − 2(10)(13) cos 72°
   a² = 269 − 260 × 0.30902
   a² = 269 − 80.34 = 188.66
   a = √188.66 ≈ 13.7 cm
5. Show Solution
   Longest side is 14 cm, so find angle C opposite it.
   cos C = (7² + 10² − 14²) / (2 × 7 × 10)
   cos C = (49 + 100 − 196) / 140 = −47/140 = −0.33571
   C = cos⁻¹(−0.33571) ≈ 109.6°
   Since C > 90°, the triangle is obtuse.
6. Show Solution
   Triangle ABC (sides 9, 15, 12): s = 18
   Area_ABC = √(18 × 9 × 3 × 6) = √2916 = 54 m²
   
   Triangle ACD (sides 7, 10, 12): s = 14.5
   Area_ACD = √(14.5 × 7.5 × 4.5 × 2.5) = √1224.84 ≈ 35.0 m²
   
   Total area = 54 + 35.0 = 89.0 m²
7. Show Solution
   We know side c = 20, angle C = 110°, and side a = 12. We want angle A.
   Using the sine rule (since C is obtuse, A must be acute, so no ambiguity):
   sin A / a = sin C / c
   sin A = 12 × sin 110° / 20 = 12 × 0.93969 / 20 = 0.56382
   A = sin⁻¹(0.56382) ≈ 34.3°
   (The sine rule is actually fine to use here since C is obtuse, guaranteeing A is acute.)
8. Show Solution
   Angle at P between bearings = 150° − 50° = 100°
   Using cosine rule with sides PA = 20, PB = 15, included angle = 100°:
   AB² = 20² + 15² − 2(20)(15) cos 100°
   AB² = 400 + 225 − 600 × (−0.17365)
   AB² = 625 + 104.19 = 729.19
   AB = √729.19 ≈ 27.0 km
9. Show Solution
   Largest angle C opposite side c = 210 m:
   cos C = (120² + 180² − 210²) / (2 × 120 × 180)
   cos C = (14400 + 32400 − 44100) / 43200
   cos C = 2700 / 43200 = 0.0625
   C = cos⁻¹(0.0625) ≈ 86.4°
   
   Area (Heron's): s = (120+180+210)/2 = 255
   Area = √(255 × 135 × 75 × 45)
   = √(255 × 135 × 75 × 45) = √116,071,875 ≈ 10,774 m²
10. Show Solution
    Area = ½ab sin C ⇒ 30 = ½(8)(10) sin C = 40 sin C
    sin C = 0.75
    C₁ = sin⁻¹(0.75) ≈ 48.6°   C₂ = 180° − 48.6° = 131.4°
    
    Both give the same area because sin(48.6°) = sin(131.4°) = 0.75.
    
    For C₁ = 48.6°:
    c² = 64 + 100 − 160 cos 48.6° = 164 − 160(0.6613) = 164 − 105.8 = 58.2
    c₁ ≈ 7.6 cm
    
    For C₂ = 131.4°:
    c² = 64 + 100 − 160 cos 131.4° = 164 + 105.8 = 269.8
    c₂ ≈ 16.4 cm