A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. Here we present a new class of optimal three-weight cyclic codes of length $q+1$ and dimension 3 over any finite field $F_q$, and show that the nonzero weights are $q-1$, $q$, and $q+1$. We then study the dual codes in this new class, and show that they are also optimal cyclic codes of length $q+1$, dimension $q-2$, and minimum Hamming distance $4$. Lastly, as an application of the Krawtchouck polynomials, we obtain the weight distribution of the dual codes.