Trigonometric functions serving as boundary functions are excellent nonlinear elements in designing chaotic maps. However, research on the dynamical behaviors dependent on the internal frequency within these boundary functions is not yet sufficient. Hence, in this paper, a novel chaotic map is proposed. Numerical simulations reveal the unique dynamical behaviors dependent on its dominant and recessive internal frequencies, including the control of the map's Lyapunov exponents and their impact on the overall system performance. This unique phenomenon has not been reported before. The system's initial boosting behavior is then captured, further revealing its initial-boosted extreme multistability. This map is implemented on the STM32 platform, demonstrating its practical applicability for potential practical application scenarios. Ultimately, the map is applied in designing a pseudo-random number generator, and its high randomness is validated through the NIST SP800-22 test.