1、第1课时两角差的余弦公式课后训练巩固提升一、A组1.化简sin(x+y)sin(x-y)+cos(x+y)cos(x-y)的结果是()A.sin 2xB.cos 2yC.-cos 2xD.-cos 2y解析:原式=cos(x+y)cos(x-y)+sin(x+y)sin(x-y)=cos(x+y)-(x-y)=cos 2y.答案:B2.已知sin =,cos(+)=-,且,则cos 等于()A.-1B.C.-D.解析:因为sin =,cos(+)=-,且,所以cos =,sin(+)=.所以cos =cos(+)-=cos(+)cos +sin(+)sin =,故选D.答案:D3.(多选题)已
2、知函数f(x)=coscos x+sinsin x,则下列结论正确的是()A.函数f(x)的最小正周期为B.函数f(x)的图象关于直线x=-对称C.函数f(x)的图象关于直线x=对称D.函数f(x)的图象关于点对称解析:因为f(x)=coscos x+sin-xsin x=coscos x+sinsin x=cos2x-,所以f(x)的最小正周期为,故A正确.由2x-=k,kZ,得x=,kZ,故f(x)的图象关于直线x=,kZ对称,故B,C正确.由2x-=k+,kZ,得x=,kZ,故f(x)的图象关于点,kZ对称,故D错误.答案:ABC4.已知cos,0,则cos 等于()A.B.C.D.解析
3、:,+.sin,cos =cos=coscos+sinsin.答案:A5.sin 162sin 78-cos 18cos 102=.解析:sin 162sin 78-cos 18cos 102=sin 18sin 78+cos 18cos 78=cos(78-18)=cos 60=.答案:6.已知钝角,满足sin =,cos =,则cos(-)=.解析:因为,是钝角,所以cos =-,sin =.所以cos(-)=cos cos +sin sin =-=-.答案:-7.在平面直角坐标系Oxy中,角与角均以Ox为始边,它们的终边关于x轴对称.若sin =,则cos(-)=.解析:角与角均以Ox为
4、始边,它们的终边关于x轴对称,sin =-sin =,cos =cos =.cos(-)=cos cos +sin sin =.答案:8.已知角的顶点为坐标原点,始边与x轴的非负半轴重合,终边经过点P(1,t),且点P到原点的距离为.(1)求实数t的值;(2)若,均为锐角,cos(+)=,求cos 的值.解:(1)由题意得1+t2=,解得t=.(2)为锐角,t=,即点P.sin =.cos =.又,为锐角,+(0,).由cos(+)=,得sin(+)=.cos =cos(+)-=cos(+)cos +sin(+)sin =.9.已知sin(3-)=sin(R),求cos的值.解:因为sin(3
5、-)=sin ,sin=cos ,所以sin =cos .因为sin2+cos2=1,所以所以coscos +sin =.二、B组1.已知sin(30+)=,60150,则cos =()A.B.C.D.解析:60150,90+30180.sin(30+)=,cos(30+)=-=-.cos =cos(30+)-30=cos(30+)cos 30+sin(30+)sin 30=-,故选A.答案:A2.已知sin x+cos x=,则cos=()A.-B.C.-D.解析:sin x+cos x=2=2=2cos,cos.答案:B3.周髀算经中给出了弦图,所谓弦图是由四个全等的直角三角形和中间一个小
6、正方形拼成一个大的正方形.若图中直角三角形两锐角分别为,且小正方形与大正方形面积之比为925,则cos(-)的值为()A.B.C.D.解析:设大的正方形的边长为1.因为小正方形与大正方形面积之比为925,所以小正方形的边长为.所以cos -sin =,sin -cos =.因为+=,所以cos =sin ,sin =cos .所以可得=cos sin +sin cos -cos cos -sin sin =sin2+cos2-cos(-)=1-cos(-),解得cos(-)=.答案:D4.若sin sin =1,则cos(-)的值为.解析:sin sin =1,|sin |1,|sin |1,
7、cos2+sin2=1,cos =0.cos(-)=cos cos +sin sin =0+1=1.答案:15.如图,在平面直角坐标系Oxy中,锐角和钝角的终边分别与单位圆相交于A,B两点.(1)如果A,B两点的纵坐标分别为,求cos 和sin ;(2)在(1)的条件下,求cos(-)的值.解:(1)OA=1,OB=1,且A,B两点的纵坐标分别为,sin =,sin =.cos =.(2)为钝角,sin =,cos =-.cos(-)=cos cos +sin sin =-.6.已知cos(-)=-,cos(+)=,且-,+,求角的值.解:-,且cos(-)=-,sin(-)=.又+,且cos(+)=,sin(+)=-.cos 2=cos(+)-(-)=cos(+)cos(-)+sin(+)sin(-)=-1.又+,-,2.2=,即=.
