Question

已知cos(4分之π-α)=5分之3,sin(4分之5π+β)=-13分之12,α∈（4分之π,4分之3π）,β∈（0,

Answer 1

∵α∈(π/4,3π/4),∴α-π/4∈(0,π/2)

cos(4分之π-α)=3/5,即cos(α-π/4)=3/5,∴sin(α-π/4)=4/5

∵β∈(0,π/4),∴β+π/4∈(π/4,π/2)

sin(4分之5π+β)=-12/13,即sin(β+π/4)=12/13,∴cos(β+π/4)=5/13

∴sin(α+β)=sin[(α-π/4)+(β+π/4)]

=sin(α-π/4)cos(β+π/4)+cos(α-π/4)sin(β+π/4)

=4/5*5/13+3/5*12/13

=56/65