求cosπ/11乘cos2π/11乘cos3π/11乘cos4π/11乘cos5π/11的值每隔1个11为一个数
问题描述：
求cosπ/11乘cos2π/11乘cos3π/11乘cos4π/11乘cos5π/11的值
每隔1个11为一个数
答
原式=(cosπ/11*cos2π/11*cos3π/11*cos4π/11*cos5π/11*2sinπ/11)/2sinπ/11=(sin2π/11*cos2π/11*cos3π/11*cos4π/11*cos5π/11)/2sinπ/11=(1/2*sin4π/11*cos3π/11*cos4π/11*cos5π/11)/2sinπ/11=(1/4*sin8π/11*cos3π/11*cos5π/11)/2sinπ/11=(1/4*sin3π/11*cos3π/11*cos5π/11)/2sinπ/11=(1/8*sin6π/11*cos5π/11)/2sinπ/11=(1/8*sin5π/11*cos5π/11)/2sinπ/11=(1/16*sin10π/11)/2sinπ/11=1/32
答
可以利用积化和差公式与半角公式,最后化cos(π/11)的表达式,
例如cos(3π/11)cos(5π/11)=1/2*[cos(4π/11)+cos(π/11)]
cos(4π/11)=2[cos(2π/11)]^2 -1
cos(2π/11)=2[cos(π/11)]^2 -1
具体过程自己做一下吧,过程繁琐点,但并不难.