In this paper, we present a systematic approach to design two families of fast multiplierless approximations of the DCT with the lifting scheme, based on two kinds of factorizations of the DCT matrix with Givens rotations. A scaled lifting structure is proposed to reduce the complexity of the transform. Analytical values of all lifting parameters are derived, from which dyadic values with different accuracies can be obtained through finite-length approximations. This enables low-cost and fast implementations with only shift and addition operations. Besides, a sensitivity analysis is developed for the scaled lifting structure, which shows that for certain rotation angles, a permuted version of it is more robust to truncation errors. Numerous approximation examples with different complexities are presented for the 8-point and 16-point DCT. As the complexity increases, more accurate approximation of the floating DCT can be obtained in terms of coding gains, frequency responses, and mean square errors of DCT coefficients. Hence the lifting-based fast transform can be easily tailored to meet the demands of different applications, making it suitable for hardware and software implementations in real-time and mobile computing applications.