第9讲三角恒等变换与解三角形1.(1)[2015·全国卷Ⅰ]已知a,b,c分别是△ABC内角A,B,C的对边,sin2B=2sinAsinC.①若a=b,求cosB;②若B=90°,且a=,求△ABC的面积.(2)[2015·全国卷Ⅱ]△ABC中,D是BC上的点,AD平分∠BAC,BD=2DC.①求;②若∠BAC=60°,求∠B.[试做]_______________________________________________________________________________________________________________________________________________________________________________________命题角度解三角形的问题(1)近五年的高考试题中,经常出现的题型有:正弦定理、余弦定理与三角变换的综合;正弦定理、余弦定理与三角形面积的综合;正弦定理、余弦定理与三角变换及三角形面积的综合.(2)解三角形问题的步骤:第一步,利用正、余弦定理进行边角转化;第二步,利用三角恒等变换求边与角;第三步,代入数据求值;第四步,转化过程中要注意转化的方向,审视结果的合理性.(3)解三角形问题的总体思路是转化思想和消元.解答1三角形基本量的求解1在△ABC中,内角A,B,C所对的边分别为a,b,c,且c-b=2bcosA.(1)若a=2,b=3,求边c的长;(2)若C=,求角B的大小.[听课笔记]__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________2在△ABC中,内角A,B,C所对的边分别是a,b,c,且2ccosB=2a-b.(1)求角C的大小;(2)当c=3时,求a+b的取值范围.[听课笔记]__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________【考场点拨】求解三角形中的边和角等基本量,需要根据正、余弦定理结合已知条件灵活转化边和角之间的关系,从而达到解决问题的目的.其基本步骤是:第一步:定条件,即确定三角形中的已知和所求,在图形中标出来,然后确定转化的方向.第二步:定工具,即根据条件和所求合理选择转化的工具,实施边角之间的互化.第三步:求结果.解答2与三角形面积有关的问题3在△ABC中,内角A,B,C所对的边分别为a,b,c,且满足asinB+bcos(B+C)=0,a=.(1)求角A的大小;(2)若b=2,求△ABC的面积.[听课笔记]_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________...
