CORDIC--based algorithms to compute cos^{-1}(t), sin^{-1}(t) and sqrt{1-t^{2}} are proposed. The implementation requires a standard CORDIC module plus a module to compute the direction of rotation, this being the same hardware required for the extended CORDIC vectoring, recently proposed by the authors. Although these functions can be obtained as a special case of this extended vectoring, the specific algorithm we propose here presents two significant improvements: (1) it achieves an angle granularity of 2^{-n} using the same datapath width as the standard CORDIC algorithm (about n bits, instead of about 2n which would be required using the extended vectoring), and (2) no repetitions of iterations are needed. The proposed algorithm is compatible with the extended vectoring and, in contrast with previous implementations, the number of iterations and the delay of each iteration are the same as for the conventional CORDIC algorithm.