已知cos（a+π/6）—sina=4/5√3,
cosacosπ/6-sinasinπ/6-sina=4√3/5,
√3cona/2-sina/2-sina=4√3/5,
5√3cona-15sina=8√3,
5√3√(1-sin^2a)=8√3+15sina,
75-75sin^2a=192+240√3sina+225sin^2a,
300sin^2a+240√3sina+117=0,
sina=[120√3±√(3*120^2-300*117)]/300=
=[120√3±√(43200-35100)]/300=
=[12√3±√71]/30,
sina=(12√3+√71)/30,或 sina=(12√3-√71)/30,
cosa=√(1-sin^2a)=√{1-[(12√3+√71)/30]^2},
或
cosa=√(1-sin^2a)=√{1-[(12√3-√71)/30]^2},
则sin（a—π/6）=sinacosπ/6-cosasinπ/6=
=(√3/2))√{1-[(12√3+√71)/30]^2}-(1/2)(12√3+√71)/30,
或者
sin（a—π/6）=sinacosπ/6-cosasinπ/6=
=(√3/2))√{1-[(12√3-√71)/30]^2}-(1/2)(12√3-√71)/30.