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Simple but accurate periodic solutions for the nonlinear pendulum equation

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Figure 1: The simple pendulum motion. At t = 0, it is released at rest from a position that forms an angle 0 < θ 0 < π rad with the vertical, then passing by an angle θ, for some 0 < t < T /4, with a scalar velocity v = ℓ θ.˙
Figure 2: Periodic approximations for the exact solution θ(t) of the pendulum equation, for θ 0 = π/3 rad
Figure 4: The same of the previous figures, for θ 0 = 7 8 π rad. The Fourier approximation was taken with N = 4

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