Catalyzes the desaturation of oleic acid (18:1(9Z)) to linoleic acid (18:2(9Z,12Z)). (9Z)-octadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-octadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O (9Z)-hexadecenoyl-CoA + 2 Fe(II)-[cytochrome b5] + 2 H(+) + O2 = (9Z,12Z)-hexadecadienoyl-CoA + 2 Fe(III)-[cytochrome b5] + 2 H2O Lipid metabolism; polyunsaturated fatty acid biosynthesis. Expressed in leaves and seeds. The histidine box domains may contain the active site and/or be involved in metal ion binding. Belongs to the fatty acid desaturase type 1 family.