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Whereas considerable evidence exists for the conversion of singlet Cooper pairs into triplet Cooper pairs in the presence of inhomogeneous magnetic fields, recent theoretical proposals have suggested an alternative way to exert control over triplet generation: intrinsic spin-orbit coupling in a homogeneous ferromagnet coupled to a superconductor. Here, we proximity couple Nb to an asymmetric Pt/Co/Pt trilayer, which acts as an effective spin-orbit-coupled ferromagnet owing to structural inversion asymmetry. Unconventional modulation of the superconducting critical temperature as a function of in-plane and out-of-plane applied magnetic fields suggests the presence of triplets that can be controlled by the magnetic orientation of a single homogeneous ferromagnet. Our studies demonstrate an active role of spin-orbit coupling in controlling the triplets, an important step towards the realization of novel superconducting spintronic devices.