1、-1-3 3.3 3三角函数的积化和差与和差化积三角函数的积化和差与和差化积-2-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航1.理解三角函数的积化和差与和差化积公式的推导过程.2.能利用积化和差与和差化积公式进行简单的三角函数式的化简、求值和证明.-3-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIA
2、ODAOHANG目标导航12-4-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12答案:B-5-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12【做一做1-2】sin 37.5cos 7.5=.-6-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONG
3、NAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12-7-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12名师点拨名师点拨1.积化和差公式的推导用了“解方程组”的思想,和差化积公式的推导用了“换元”思想.2.不论是积化和差还是和差化积中的“和差”与“积”,都是指三角函数间的关系,并不是指角的关系.-8-3.3三角函数的积化和差与和差化积ZHISHI SH
4、ULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12答案:C-9-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航12-10-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航1.
5、和差化积与积化和差公式的作用剖析(1)可从以下几方面来理解这两组公式:这些公式中的“和差”、“积”都是指三角函数间的关系,并不是指角的关系;只有系数绝对值相同的同名三角函数的和差,才能直接应用公式化为积的形式.如sin+cos 就不能直接化积,应先化成同名函数后,再用公式化成积的形式;三角函数的和差化积,可以理解为代数中的因式分解.(2)一般情况下,遇有正弦函数、余弦函数的平方,要先考虑灵活应用二倍角公式的变形进行降幂,然后应用和差化积、积化和差公式进行化简或计算.-11-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYA
6、NLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航(3)和差化积、积化和差公式的基本功能在于:当和积互化时,角度要重新组合,因此有可能产生特殊角;结构将变化,因此有可能产生互消项或互约因式,从而利于化简求值.正因为如此,“和积互化”是三角恒等变形的一种基本方法.在解题过程中,当遇到三角函数的和时,就试着化为积的形式;当遇到三角函数的积时,就试着化为和差的形式.往往这样就能发现解决三角函数问题的思路.为了能够把三角函数化成积的形式,有时需要把某些数当作三角函数值,-12-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJI
7、AO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航-13-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航-14-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航-15-3.3三角函数的积化和差与和差化积ZHISHI S
8、HULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五分析解答本题,先利用积化和差、和差化积公式对所求式子进行变形,再利用特殊角的三角函数值或所给条件求解.-16-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五-17-3.3三角函数的积化和差与和差化积ZHISHI SHULI知
9、识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五反思反思通过和差化积与积化和差公式的运用,一是可产生特殊角,二是虽不是特殊角,但可正、负相消或分子、分母约分,从而达到求值的目的.-18-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五【变式训练1】求下列各式的值:(2)cos 146
10、+cos 94+2cos 47cos 73.-19-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五【例2】化简:4sin(60-)sin sin(60+).分析观察(60-)与(60+)的和为特殊角,所以可用积化和差公式化简.解:原式=-2sin cos 120-cos(-2)=sin+2sin cos 2=sin+(sin 3-sin)=sin 3.反思反思此题直接考查公式的应用,对于这种题目,解题公式的
11、选取是关键.-20-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五-21-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五【例3】在ABC中,求证:sin2A+sin2B-sin2C=2sin Asin Bcos C.分析
12、先用降幂公式,再利用和差化积公式.=cos2C-cos(A+B)cos(A-B)=cos Ccos(A-B)-cos(A+B)=2sin Asin Bcos C=右边.故原式成立.反思反思在三角形中证明恒等式时,一方面要充分利用内角和为这一条件,另一方面要注意降幂公式与和差化积、积化和差公式的合理运用.-22-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五【变式训练3】求证:sin2-sin2=sin(+)
13、sin(-).证明左边=sin2-sin2=(sin+sin)(sin-sin)=sin(+)sin(-)=右边,所以等式成立.-23-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五分析将cos2A+cos2B利用降幂公式与积化和差、和差化积公式化为正弦形式或余弦形式.-24-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN
14、随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五-25-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五反思反思求一个三角函数式的单调性、最值、周期或值域等,一般要先将函数式化简为类似Asin(x+)+k的形式,再进行求解.对于本题,不要错误地求解为:cos2A和cos2B的最大值均为1,最小值都是0,所以原式的最大值为2,最小值为0.-26-3.3三角
15、函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五答案:C-27-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五错解:根据和差化积公式的结构形式,选项D中应该有负号,故选D.错因分析忽视了和差化积公式的内在关系,只看到了表面现象,没注
16、意到角的顺序也有变化.-28-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五答案:C -29-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航题型一题型二题型三题型四题型五【变式训练5】求值:sin 55-sin 65+sin 5.解:sin 55-s
17、in 65+sin 5=+sin 5=2cos 60sin(-5)+sin 5=-sin 5+sin 5=0.-30-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航1234561.有下列关系式:sin 5+sin 3=2sin 8cos 2;cos 3-cos 5=-2sin 4sin;sin 3-sin 5=-cos 4cos;sin 5+cos 3=2sin 4cos;sin xsin y=cos(x-y)-cos(x+y).其中正
18、确等式的个数是()A.0B.1C.2D.3解析:均不正确,正确.答案:B-31-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航1234562.cos 40+cos 80+cos 160的值等于()C.cos 20 D.2cos 20解析:原式=2cos 60cos(-20)+cos 160=cos 20+cos 160=0.答案:B-32-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重
19、难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航123456答案:D-33-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航123456-34-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航123456-35-3.3三角函数的积化和差与和差化积ZHISHI SHULI知识梳理ZHONGNAN JVJIAO重难聚焦SUITANGYANLIAN随堂演练DIANLI TOUXI典例透析MUBIAODAOHANG目标导航1234566.求sin220+cos250+sin 20cos 50的值.
