若锐角α、β满足(1+根号三tanα)(1+根号三tanβ)=4,求α+β的值
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若锐角α、β满足(1+根号三tanα)(1+根号三tanβ)=4,求α+β的值
若锐角α、β满足(1+根号三tanα)(1+根号三tanβ)=4,求α+β的值
若锐角α、β满足(1+根号三tanα)(1+根号三tanβ)=4,求α+β的值
由(1+根号三tanα)(1+根号三tanβ)=4
得 1+根号3倍(tanα+tanβ)+3tanαtanβ=4
tanα+tanβ=(3-3tanαtanβ)/根号3
=根号3(1-tanαtanβ)
( tanα+tanβ)/(1-tanαtanβ)=根号3
即 tan(α+β)=根号3
又∵ α、β为锐角
∴ α+β=π/3
(1+√3tanα)(1+√3tanβ)=4,1+√3(tanα+tanβ)+3tanαtanβ=4
√3(tanα+tanβ)+3tanαtanβ=3,(tanα+tanβ)+√3tanαtanβ=√3
tanα+tanβ=√3(1-tanαtanβ)
(tanα+tanβ)/(1-tanαtanβ)=tan(α+β)=√3,α+β=π/3
π/3