Question

高一数学 详细解释一下

求值cosπ/17*cos2π/17*cos3π/17*cos4π/17*cos8π/17
题目错了，应是
求值cosπ/17*cos2π/17*cos4π/17*cos8π/17

Answer 1

cosπ/17*cos2π/17*cos4π/17*cos8π/17
=(2^4sinπ/17*cosπ/17*cos2π/17*cos4π/17*cos8π/17)/2^4sinπ/17
=(2^3sin2π/17*cos2π/17*cos4π/17*cos8π/17)/2^4sinπ/17
=(2^2sin4π/17*cos4π/17*cos8π/17)/2^4sinπ/17
=(2sin8π/17*cos8π/17)/2^4sinπ/17
=sin16π/17/2^4sinπ/17
=sin(π-π/17)/2^4sinπ/17
=sinπ/17/2^4sinπ/17
=1/16

Answer 2

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Answer 3

式子左边乘以sinπ/17，则式子变成1/2*sin2π/17*cos2π/17*cos4π/17*cos8π/17，以此类推，最后变成1/16*sin(16π/17)，而sin(16π/17)=sinπ/17，式子开始乘以了sinπ/17，所以，结果1/16*sin(16π/17)要处以sinπ/17，即最后答案为1/16。