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www.qqv8.com:江苏省宿迁市高中数学第3课时余弦定理导学案(无答案)苏教版必修5

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内部文件,版权追溯 第 3 课时 【学习目标】 余弦定理 1.掌握余弦定理,并能解决一些简单的三角形度量问题; 2.通过解三角形边培养学生的综合运用知识的能力,和计算能力. 【问题情境】 1. ,通过向量的数量积即: BC ? AD ? ( BA ? AC) ? AD( AD为?ABC中BC边上的高) , 进而推出了正弦定理。还有其他途径将向量等式 BC = BA ? AC 数量化吗? 2. 是否还有其它的向量等式?类比推导还可以得到的等式有哪些. 【合作探究】 1.探究一 等式的结构有何特点,余弦定理可以如何变形. 2.探究二 利用余弦定理,可以解决的解三角形问题. 3.知识建构 (1)余弦定理_____________________________________________. (2)利用余弦定理,可以解决的解三角形问题 _______________________________________________________________________ _______________________________________________________________________ 【展示点拨】 例1. 在△ABC 中, 1 (1)已知 AB ? 2, AC ? 3, A ? 60? ,求 BC ; (2)已知 a ? 20, b ? 29, c ? 21,求 B ; (3)如果 sin A : sin B : sin C ? 2 : 3 : 4 ,求 cos C . 例 2.如图,在△ABC 中,已知 AB=4, AC=3,角平分线 AD=2, 求 A △ABC 的面积. B D C 例 3. 用余弦定理证明:在△ABC 中,当∠C为锐角时, a ? b ? c ;当∠C为钝角时, 2 2 2 a 2 ? b2 ? c 2 . 例 4. 如图,AM 是△ABC 中 BC 边上的中线, 求证: AM ? 1 2( AB 2 ? AC 2 ) ? BC 2 . 2 2 拓展延伸:书 P17:第 8 题 【学以致用】 1. 在△ABC 中,已知 (a ? b ? c)(b ? c ? a) ? 3bc ,则 A 度数为__________________. 2.在△ABC 中,AB=2,BC=5,且△ABC 的面积为 3,则 AC=___________. 3.在△ABC 中, a ? 5, b ? 6, c ? 8 ,则△ABC 是————三角形. 0 4.在△ABC 中, AB ? AC ? 2 , BC ? 2 3 ,点 D 在 BC 边上, ?ADC ? 45 ,则 AD 的长 度等于_____. 第 3 课时 余弦定理(1)同步训练 【基础训练】 1.在△ABC 中,已知 b=1,c=3,A=60°,则 a = 2. 在△ABC 中,已知 a =20,b=29,c=21,则 B= 3. 在△ABC 中,已知 a =4,b=5,c=6,则 cosA= 4. 在△ABC 中,已知 a =5,c=4,A=60°,则 b= . . . . 3 5. 在△ABC 中,若边长 a ,b,c 满足 1 1 3 ,则角 C= ? ? b?c c?a a?b?c . . 6. 在△ABC 中,已知 b=3,则 a cos C ? c cos A = 2 7. 在 Δ ABC 中 , A=60 ? , 最 大 边 与 最 小 边 是 方 程 3x ? 27 x ? 32 ? 0 的 两 根 , 则 BC= . . 8.钝角三角形的三边长为连续自然数,则这三边长分别为 【思考应用】 9. 在△ABC 中,已知 a =2,b= 2 2 ,C=15°,求 A. 10. 在锐角△ABC 中,b=1,c=2,求 a 的取值范围. 4 【拓展提升】 11. 如图,在 ?ABC 中,AC=2,BC=1, cos C ? 值. 3 . (1)求 AB 的值; (2)求 sin ?2 A ? C ? 的 4 12. 在△ABC 中, 三边长为连续的自然数, 且最大角是最小角的 2 倍, 求此三角形的三边长. 5

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