Question

已知α,β均为锐角,且tan(α+π/4)=3,sin(α-β)=-√10/10,则tanβ等于

已知α,β均为锐角,且tan(α+π/4)=3,sin(α-β)=-√10/10,则tanβ等于

Answer 1

0<α<π/2, 0< β<π/2,
则-π/2<α-β<π/2，cos(α-β)>0.
sin(α-β)=-√10/10,
cos(α-β)=√(1- sin²(α-β))= 3√10/10,
所以tan(α-β)=-1/3.

tan(α+π/4)=3,
即(tanα+ tanπ/4)/(1- tanαtanπ/4) =3,
∴tanα=1/2.

tanβ=tan[α-(α-β)]
=(tanα- tan(α-β))/(1+ tanαtan(α-β))
=1.