Question

Minimum vertical distance between graph of `y=2+sinx and y=cosx` is ?

Answer 1

Subtracting the second function from first we get the resultant function representing the variable vertical distance between the graphs with variation of `x`

So let it be

`D(x)=2+sinx-cosx`

`=>D(x)=2+sqrt2(sinx*1/sqrt2-cosx*1/sqrt2)`

`=>D(x)=2+sqrt2(sinxcos(pi/4)-cosxsin(pi/4))`

`=>D(x)=2+sqrt2sin(x-pi/4)`

`D(x)` will be minimum when `sin(x-pi/4)=-1`,the minimum value of sine function.

Hence

`[D(x)]_"min"= 2-sqrt2`