若tanx=2tanπ尀5，则cos(x-3尀10π)尀sin(x-π尀5)=

本题主要考查对 三角函数的诱导公式 考点的理解。解题过程如下：

tanx=2tanπ/5
cos(x-3π/10)/sin(x-π/5)
={cosxcos3π/10+sinxsin3π/10) / (sinxcosπ/5-cosxsinπ/5)
（分子分母同除以cosx）
={cos3π/10+tanxsin3π/10) / (tanxcosπ/5-sinπ/5)
={cos3π/10+2tanπ/5sin3π/10) / (2tanπ/5cosπ/5-sinπ/5)
(分子分母同乘以cosπ/5）
={cosπ/5cos3π/10+2sinπ/5sin3π/10) / (2sinπ/5cosπ/5-cosπ/5sinπ/5)
={cosπ/5cos3π/10+sinπ/5sin3π/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cos(3π/10-π/5)+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10-1/2[cos(π/5+3π/10)-cos(π/5-3π/10)]/ sinπ/5cosπ/5
={cosπ/10-1/2[cosπ/2-cos(π/10)]/ sinπ/5cosπ/5
= [(3/2)cosπ/10] /[1/2sin2π/5]
= [3cosπ/10] /[sin2π/5]
= [3cosπ/10] /[cos(π/2-2π/5)]
= [3cosπ/10]/[cosπ/10]
= 3

诱导公式：

规律：奇变偶不变，符号看象限。即形如（2k+1）90°±α，则函数名称变为余名函数，正弦变余弦，余弦变正弦，正切变余切，余切变正切。形如2k×90°±α，则函数名称不变。

tanx=2tanπ/5
cos(x-3π/10)/sin(x-π/5)
={cosxcos3π/10+sinxsin3π/10) / (sinxcosπ/5-cosxsinπ/5)
（分子分母同除以cosx）
={cos3π/10+tanxsin3π/10) / (tanxcosπ/5-sinπ/5)
={cos3π/10+2tanπ/5sin3π/10) / (2tanπ/5cosπ/5-sinπ/5)
(分子分母同乘以cosπ/5）
={cosπ/5cos3π/10+2sinπ/5sin3π/10) / (2sinπ/5cosπ/5-cosπ/5sinπ/5)
={cosπ/5cos3π/10+sinπ/5sin3π/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cos(3π/10-π/5)+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10-1/2[cos(π/5+3π/10)-cos(π/5-3π/10)]/ sinπ/5cosπ/5
={cosπ/10-1/2[cosπ/2-cos(π/10)]/ sinπ/5cosπ/5
= [(3/2)cosπ/10] /[1/2sin2π/5]
= [3cosπ/10] /[sin2π/5]
= [3cosπ/10] /[cos(π/2-2π/5)]
= [3cosπ/10]/[cosπ/10]
= 3

您好：tanx=2tanπ/5
cos(x-3π/10)/sin(x-π/5)
={cosxcos3π/10+sinxsin3π/10) / (sinxcosπ/5-cosxsinπ/5)
（分子分母同除以cosx）
={cos3π/10+tanxsin3π/10) / (tanxcosπ/5-sinπ/5)
={cos3π/10+2tanπ/5sin3π/10) / (2tanπ/5cosπ/5-sinπ/5)
(分子分母同乘以cosπ/5）
={cosπ/5cos3π/10+2sinπ/5sin3π/10) / (2sinπ/5cosπ/5-cosπ/5sinπ/5)
={cosπ/5cos3π/10+sinπ/5sin3π/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cos(3π/10-π/5)+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10+sinπ/5sin3π/10) / sinπ/5cosπ/5
={cosπ/10-1/2[cos(π/5+3π/10)-cos(π/5-3π/10)]/ sinπ/5cosπ/5
={cosπ/10-1/2[cosπ/2-cos(π/10)]/ sinπ/5cosπ/5
= [(3/2)cosπ/10] /[1/2sin2π/5]
= [3cosπ/10] /[sin2π/5]
= [3cosπ/10] /[cos(π/2-2π/5)]
= [3cosπ/10]/[cosπ/10]
= 3

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