該題公式由清代數學家李善蘭先生提出,後稱輔助角公式。其推導如下:
由題意得asinx+bcosx
令c=a²+b²
則asinx+bcosx
=√c[(a/√c)sinx+(b/√c)cosx]
設cosy=a/√c, siny=b/√c,即tany=a/b,則y=arctan(a/b)
故asinx+bcosx
=√c(sinxcosy+cosxsiny)
=√csin(x+y)
=√(a²+b²)sin[x+arctan(a/b)]