求值cos^2a+cos^2(a+120°)+cos^2(a+240°)

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/20 20:23:07
求值cos^2a+cos^2(a+120°)+cos^2(a+240°)

求值cos^2a+cos^2(a+120°)+cos^2(a+240°)
求值cos^2a+cos^2(a+120°)+cos^2(a+240°)

求值cos^2a+cos^2(a+120°)+cos^2(a+240°)
cos^2a+cos^2(a+120°)+cos^2(a+240°)
=(cosa)^2+{-cos[180-(a+120)]}^2+{-cos[180-(a+240)]}^2
=(cosa)^2+[cos(60-a)]^2+[cos(60+a)]^2
=(cosa)^2+(cos60cosa+sin60sina)^2+(cos60cosa-sin60sina)^2
=(cosa)^2+(1/2*cosa+√3/2*sina)^2+(1/2*cosa-√3/2*sina)^2
=(cosa)^2+1/4(cosa)^2+√3/2sinacosa+3/4(sina)^2+1/4(cosa)^2-√3/2sinacosa+3/4(sina)^2
=3/2*(cosa)^2+3/2*(sina)^2
=3/2*[(sina)^2+(cosa)^2]
=3/2