已知3tan尔发=2tan(尔发+贝塔),求sin(2尔发+贝塔)/sin贝塔

A=尔发，B=贝塔
3tanA=2tan(A+B)
tanA/tan(A+B)=2/3
sin(2A+B)/sinB=sin(A+(A+B))/sin((A+B)-A)
=[sinAcos(A+B)+cosAsin(A+B)]/[sin(A+B)cosA-cos(A+B)sinA]
(分子分母同时除以cosAcos(A+B)
=(tanA+tan(A+B))/(tan(A+B)-tanA)
分子分母同时除以tan(A+B)
=(1+tanA/tan(A+B))/(1-tanA/tan(A+B))=(1+2/3)/(1-2/3)=5

3tanα=2tan(α+β),求sin(2α+β)/sinβ
解：sin(2α+β)/sinβ
=(sin2αcosβ+cos2αsinβ)/sinβ
=[2sinαcos²β+(cos²α-sin²α)sinβ]/sinβ
=2sinαcos²β/sinβ+cos²α-sin²α
因为 3tanα=2tan(α+β)
所以 3sinα/cosα=2(tanα+tanβ)/(1-tanαtanβ)
3sinα/cosα=2(sinα/cosα+sinβ/cosβ)/[1-(sinα/cosα)(sinβ/cosβ)]