3tan(a-π/12)=tan(a+π/12) 求a

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3tan(a-π/12)=tan(a+π/12) 求a
3tan(a-π/12)=tan(a+π/12) 求a

3tan(a-π/12)=tan(a+π/12) 求a
3tan(a-π/12)=tan(a+π/12)
3sin(a-π/12)cos(a+π/12)=sin(a+π/12)cos(a-π/12)
3[sin2a-sin(π/6)]=sin2a+sin(π/6)
2sin2a=4sin(π/6)=4*(1/2)=2
sin2a=1
2a=2kπ+π/2
a=kπ+π/4 (k∈Z)

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你令a-π/12=x则a+π/12=x+π/6
然后运用正切值的求和公式即可求解。

设M=tan(a-π/12) tan(a+π/12)=3M
tan[(a-π/12)-(a+π/12)]=-tan(π/6)
-tan(π/6)=-√3/3=[tan(a-π/12)-tan(a+π/12)]/[1+tan(a-π/12)tan(a+π/12)]
-√3/3=(M-3M)/(1+3M^2)
3M^2+2√3M+1=0
(√3M+1)^2=0
M=-√3/3
tan(a-π/12)=tan(-π/6) 或tan(a-π/12)=tan(5π/6)
a=-π/12+kπ

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