Approximate frequencies of the pendulum for large angles

By approximating the cosine function to a polynomial, analytical approximations of pendulum trajectories are developed for initial angles close to $\pi $. The periods are deduced and they are compared with other techniques recently developed for the same purpose. Our results practically match with the exact solutions. A process that allows to understand the difficulties of dealing with nonlinear equations, of using the minimization of the standard deviation and the importance played by energy conservation is done.