Question

高二数学 数列

sin47°-sin11°+sin61°-sin25°-cos7°的值是（ ）

Answer 1

sin47°+sin61°-sin11°-sin25°

=(sin47°+sin61°)-(sin11°+sin25°)

=sin〔1/2*(47+61)+1/2*(47-61)〕+sin〔1/2*(47+61)-1/2*(47-61)〕-{sin〔1/2*(25+11)+1/2*(25-11)〕+sin〔1/2*(25+11)-1/2*(25-11)〕}

=sin(54+7)+sin(54-7)-〔sin(18+7)+sin(18-7)〕

=sin54cos7+cos54sin7+sin54cos7-cos54sin7-〔sin18cos7+cos18sin7+sin18cos7-cos18sin7〕
= 2sin54°*cos7°-2sin18°*cos7°

=2cos7°*(sin54°+sin(-18°))

=2cos7°*2sin18°*cos36°

=4 cos7°*sin18°*cos18°*cos36°/cos18°

=2cos7°*sin36°*cos36°/cos18°

=cos7°*sin72°/cos18°

=cos7°

所以sin47°-sin11°+sin61°-sin25°-cos7°=0