Locally recoverable codes are very important due to their applications in distributed storage systems. In this paper, by using cyclic codes, we construct two classes of optimal q-ary cyclic (r, δ1) locally recoverable codes with parameters [2(q + 1), 2(q + 1) - 2δ1, δ1+ 2]q, where q is an odd prime power and r + δ1- 1 = q + 1, and (r, δ2) locally recoverable codes with parameters [2(q + 1), 2(q + 1) - 4δ2+ 2, δ2+ 2]q, where q ≥ 7 is an odd prime power, q ≡ 3 (mod 4) and r + δ2- 1 = q+1 2 . Compared with the known cyclic and constacyclic (r, δ) locally recoverable codes, our construction yields new optimal cyclic (r, δ) locally recoverable codes.