Question

f(x) 的图像按向量a=(π/6,-2)平移，得到函数g（x)图像

Answer 1

f(x)=2√3cosx-2sinx) / (5+2cos²x-2√3sinxcosx) +2
=4(√3/2*cosx-1/2*sinx) / [5+ 4cosx(1/2*cosx-√3/2*sinx)] +2
=4[cos(pi/6)*cosx-sin(pi/6)*sinx] / [5+ 4cosx(sin(pi/6)*cosx-cos(pi/6)*sinx)] +2
=4cos(pi/6+x) / [5+4cosx* sin(pi/6-x)] +2

f(x)按向量(pi/6,-2)平移后的函数是
g(x)=f(x-pi/6)-2
=4cosx / [5+4cos(x-pi/6)* sin(pi/3-x)]
=4cosx / {5+4[cosx*cos(pi/6)+sinx*sin(pi/6)]*[sin(pi/3)cosx-cos(pi/3)sinx]}
=4cosx / [5+4(√3/2*cosx + 1/2*sinx)*(√3/2*cosx - 1/2*sinx)]
=4cosx / [5+4(3/4*cos²x - 1/4*sin²x)]
=4cosx / (5+3cos²x - sin²x)
=4cosx / (4+3cos²x +1-sin²x)
=4cosx / (4+4cos²x)
=cosx / (1+cos²x)