1.

Find the inverse of the function `f:[-(pi)/(2)-"tan"^(_1)(3)/(2),(pi)/(2)-"tan"^(-1)(3)/(4)] to [-1,1],` `f(x)=3cosx+4sinx +7.`

Answer» We have,
`f(x)= y=3cosx +4sinx+7`
`implies y=sqrt(3^(2)+4^(2))[(3)/(sqrt(3^(2)+4^(2)))cosx+(4)/(sqrt(3^(2)+4^(2)))sinx]+7`
`=5[(3)/(5)cosx+(4)/(5)sinx]+7`
`=5[sin theta cos x+cos theta sinx]+7," where " tan theta=(3)/(4)`
`=5 sin(x+theta)+7`
`implies y-7=5sin(x+theta)`
`implies (y-7)/(5)=sin(x+theta)`
`implies "sin"^(-1)(y-7)/(5)=x+theta`
`implies x="sin"^(-1)(y-7)/(5)-theta`
`f^(-1)(x)="sin"^(-1)(x-7)/(5)-"tan"^(-1)(3)/(4)`


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