练案5 10.1.1 复数的概念-【成才之路】2021-2022学年高中新教材数学必修第四册新课程同步学习指导（人教B版）

▲ 159 ▲ ▲ 160 ▲ 故 BD = AB·sin A = 3 × 32 = 3 3 2 . 3. D　 ∵ AB→·AC→ = |AB→ |· |AC→ |·cos 〈AB→,AC→〉, 由向量模的定义和余弦定理可以得出 |AB→ | = 3, |AC→ | = 2, cos 〈AB→,AC→〉 = AB 2 + AC2 - BC2 2AB·AC = 1 4 . 故AB→·AC→ = 3 × 2 × 14 = 3 2 . 4. A　 cos C = 2cos 2 C2 - 1 = 2 × 5 5( ) 2 - 1 = - 35 , 在△ABC 中,由余弦定理,得 AB2 = CA2 + CB2 - 2CA·CB·cos C, 所以 AB2 = 1 + 25 - 2 × 1 × 5 × - 35( )= 32,所以 AB = 4 2 . 5. π3 　 ∵ p = (a + c,b),q = (b - a,c - a),p∥q, ∴ (a + c)(c - a) - b(b - a) = 0,即 a2 + b2 - c2 = ab. 由余弦定理,得 cos C = a 2 + b2 - c2 2ab = ab 2ab = 1 2 , ∵ 0 < C < π,∴ C = π3 . 6. 4　 因为 b + c = 7,所以 c = 7 - b. 由余弦定理得:b2 = a2 + c2 - 2accos B,即 b2 = 4 + (7 - b) 2 - 2 × 2 × (7 - b) × - 14( ),解得 b = 4. 7. (1)在△ABC 中, S△ABC = 1 2 × AB × BCsin ∠ABC,∴ 1 2 × 3 × BCsin 2π 3 = 3 3 4 , ∴ BC = 3,AB = BC. 又∵ ∠ABC = 2π3 ,∴ ∠ACB = π 6 . (2)∵ BC⊥CD,∴ ∠ACD = π3 . 由余弦定理得 AC2 = AB2 + BC2 - 2AB·BCcos 2π3 = ( 3) 2 + ( 3) 2 - 2 3 × 3 × - 12( )= 9, ∴ AC = 3. 在△ACD 中,由正弦定理得, ACsin ∠ADC = AD sin ∠ACD, ∴ AD = ACsin ∠ACDsin ∠ADC = 3sin π3 sin π4 = 32 6. 8. (1)在△ABC 中,∵ 2acos B - ccos B = bcos C, ∴ 2sin Acos B = sin Ccos B + sin Bcos C = sin(B + C) = sin A. ∵ A∈(0,π),∴ sin A≠0,则 cos B = 12 . ∵ B∈(0,π),∴ B = π3 . (2)在△ABD 中,由余弦定理,得 AD2 = 12 a( ) 2 + c2 - 2 × 12 accos B = 1 4 a 2 + c2 - 12 ac. 在△ABC 中,由余弦定理,得 b2 = a2 + c2 - 2accos B = a2 + c2 - ac. ∵ AD = b,∴ 14 a 2 + c2 - 12 ac = a 2 + c2 - ac,整理得 34 a 2 = 12 ac. ∴ ac = 2 3 ,∴ sin A sin C = a c = 2 3 . 练案[4] A 组　 素养自测 1. D　 在△ABC 中,AB = 10,BC = 20,∠ABC = 120°,则由余弦定理,得 AC2 = AB2 + BC2 - 2AB·BCcos∠ABC = 100 + 400 - 2 × 10 × 20cos 120° = 100 + 400 - 2 × 10 × 20 × - 12( )= 700, ∴ AC = 10 7,即 A、C 两地的距离为 10 7 km. 2. D　 本题中 a、c、β 这三个量不易直接测量,故选 D. 3. B　 在 Rt△ADC 中,AC = hsin β,在△ABC 中,由正弦定理,得 BC = ACsin(β - α)sin α = hsin(β - α) sin αsin β . 4. C　 在△ACD 中,∠ADC = 67. 5°,∠ACD = 45°⇒∠DAC = 67. 5°⇒AC = DC = 2 3千米; 在△BCE 中,∠BCE = 75°,∠BEC = 60° ⇒∠CBE = 45°,由正弦定理得 CE sin ∠CBE = BC sin ∠BEC⇒BC = 3千米, 在△ABC 中,∠ACB = 60°,由余弦定理得 AB2 = AC2 + BC2 - 2AC·BC·cos ∠ACB⇒AB = 3 千米. 故选 C. 5. AC　 本题考查余弦定