2017-2018学年高中数学 第三章 三角恒等变换 3.3 三角函数的积化和差与和差化积 新人教B版必修4_图文

三角函数的积化和差与和差化积 预习课本 P149~151,思考并完成以下问题 (1)如何利用两角差(和)的正、余弦公式导出积化和差与和差化 积公式? (2)两组公式有何特点? [新知初探] 1.三角函数的积化和差 cos αcos β=__12_[_c_o_s_(α__+__β_)_+__c_o_s_(α__-__β_)]_, sin αsin β=_-__12_[_c_o_s_(_α_+__β_)_-__c_o_s_(α__-__β_)_] _, sin αcos β=___12_[s_i_n_(_α_+__β_)_+__s_in__(α_-___β_)]__, cos αsin β=__12_[_s_i_n_(α__+__β_)-___si_n_(_α_-__β_)_]__. [点睛] 积化和差公式的结构特点 (1)同名函数积化为余弦函数的和差;异名函数积化为正弦函 数的和差. (2)角的顺序,“α+β”在前,“α-β”在后. 2.三角函数的和差化积 x+y x-y sin x+sin y= 2sin 2 cos 2 , x+y x-y sin x-sin y= 2cos 2 sin 2 , x+y x-y cos x+cos y= 2cos 2 cos 2 , cos x-cos y=-2sinx+2 ysinx-2 y . [点睛] 和差化积公式的特点 (1)同名函数的和或差才可化积. (2)余弦函数的和或差化为同名函数之积. (3)正弦函数的和或差化为异名函数之积. (4)等式左边为单角 α 和 β,等式右边为α-2 β与α+2 β的形式. (5)只有余弦函数的差化成积式后的符号为负,其余均为正. [小试身手] 1.下列等式错误的是 A.sin(A+B)+sin(A-B)=2sin Acos B B.sin(A+B)-sin(A-B)=2cos Asin B C.cos(A+B)+cos(A-B)=2cos Acos B D.cos(A+B)-cos(A-B)=2cos Acos B 答案:D () 2.sin 37.5°cos 7.5°等于 2+1 A. 2 2+1 C. 4 答案:C 3.cos 75°cos 15°=________. 答案:14 () 3+ 2 B. 2 3+ 2 D. 4 化简求值 [典例] 化简:4sin(60°-θ)·sin θ·sin(60°+θ). [解] 原式=2sin θ[2sin(60°-θ)·sin(60°+θ)] =-2sin θ[cos 120°-cos(-2θ)] =-2sin θ·???-12-cos ? 2θ ? ? =sin θ+2sin θ·cos 2θ =sin θ+(sin 3θ-sin θ)=sin 3θ. 用和差化积公式化简三角函数式时,若三角函数式中 存在三个或三个以上的三角函数可供化积时,应选择两角 和或差的一半是特殊角或与其他三角函数有公因式的两个 三角函数进行和差化积. [活学活用] 求 sin270°+cos240°-sin 70°cos 40°的值. 解:原式=1-co2s 140°+1+co2s 80°-sin 70°cos 40°=1+12(cos 40°+cos 80°)-sin 70°cos 40°=1+cos 60°cos 20°-12(sin 110° +sin 30°)=1+12cos 20°-12cos 20°-14=34. 三角恒等式证明 [典例] 在△ABC 中,求证:sin 2A+sin 2B+sin 2C=4sin Asin Bsin C. [证明] 左 边 = sin 2A + sin 2B + sin 2C = 2sin 2A+2B 2 cos2A-2 2B+sin 2C =2sin(A+B)cos(A-B)-2sin(A+B)cos(A+B) =2sin C[cos(A-B)-cos(A+B)] =2sin C·(-2)sin?A-B?+2 ?A+B?sin?A-B?-2 ?A+B? =4sin Asin Bsin C=右边. 所以原等式成立. 三角恒等式的证明 (1)证明三角恒等式从某种意义上来说,可以看成已知结 果的三角函数式的化简与求值. (2)证明三角恒等式总体要求是:通过三角公式进行恒等 变形,论证等式左右两边相等,论证过程要清晰、完整、推 理严密. (3)证明三角恒等式的基本思想是:化繁为简、左右归一、 变更论证等. [活学活用] 求证:cos2x+cos2(x+α)-2cos αcos xcos(x+α)=sin2α. 证明:左边=1+c2os 2x+1+cos?22x+2α?-2cos αcos x·cos(x+α) =1+12[cos 2x+cos(2x+2α)]-2cos αcos x cos(x+α) =1+cos2x+22x+2αcos2x-22x-2α-cos α[cos(2x+α)+cos α] =1+cos(2x+α)cos α-cos αcos(2x+α)-cos2α =1-cos2α=sin2α=右边, ∴原等式成立.

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