How about FrameLabel -> {"x coordinate", "y coordinate (b)"}. Otherwise you can place the sinusoid as an inset in Graphics:
sinusoid =
Plot[Sin[x], {x, 0, 4*2*\[Pi]}, Frame -> True, PlotRange -> All,
PlotStyle -> Directive[Blue, Thickness[0.004]], ImageSize -> 300,
FrameStyle -> Directive[Black, Thickness[0.003]],
LabelStyle -> Directive[Black, 10],
FrameLabel -> {"x coordinate",
Row[{"y coordinate ", Rotate[ "(b)", -Pi/2]}]}];
Graphics[{Text[Style["(b)", 10, Red], {0, 1}],
Inset[sinusoid, {.1, .95}, {0, 1}]}, PlotRange -> {{0, 1}, {0, 1}},
PlotRangePadding -> .1]
You will have to play with the parameters to get the composition you like.
