It depends on your definiton of wide.. Your 18-200 offer a angle of view (AOV) 75 degree whereas Sigma 10-20mm F4-5.6 EX DC HSM offers a 102 degree ( S$ 888) , Sigma 12-24mm F4.5-5.6 EX DG ASPHERICAL HSM offers a 122 degree (est S$ 1050) and Nikon AF-S DX 12-24mm f/4G IF-ED offers 99 degree ( est S$1630).
These 3 lens' AOV speaks for itself. Go down a shop and test it.... Happy Lunar New Year