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This article focuses on increasing the accuracy of Directional-of-Arrival (DOA) estimation by synthesizing the Conformal Uniform Circular Array (CUCA) and uses the Rank Reduction (RARE) algorithm to minimize the mutual coupling and shadowing impacts on the array performance. After separating the azimuthal angles from all of the unknown information by transforming the CUCA from the element space to the mode space, the RARE algorithm is applied in the estimated azimuthal angles. Also, the ambiguous values of π exist in the RARE algorithm are solved by forming the CUCA beam pattern. Due to the fact that the accurate measuring or estimating the mutual coupling and shadowing effects are difficult, one of the most important achievements of this work is that there is no need to measure the mutual coupling and shadow effect. Also, the CUCA aperture is not reduced compared to use the subarray methods as another advantage. The simulation results significantly confirm the presented method's benefits.