Approximate arcsinc

Haskell, 34 bytes

f m=until(\x->sin x< -m*x)(+1e-3)0

Counts x up from 0 by 0.001 until sin(x)< -m*x.

Ouput examples

f -0.2 ->   2.595999999999825
f -0.1 ->   2.852999999999797
f  0.0 ->   3.141999999999765
f  0.1 ->   3.4999999999997256
f  0.2 ->   4.1049999999997056

Mathematica, 28 bytes

x/.FindRoot[Sinc@x+#,{x,1}]&

Searches for a numerical root from the initial guess x=1. Test cases:

% /@ {-0.25, -0.1, 0.2}
(* {2.47458, 2.85234, 4.10462} *)