Large Amplitude Pendulums

Without the small-angle approximation the differential equation is not the simple harmonic motion equation and the period is not independent of the amplitude. The exact period can't be written in simple closed form, but it can be written as an elliptic integral of the first kind (which can be numerically approximated) or written as a power series expansion. Short of that, various other approximations are also available. Plot one or two of the approximations you learn about below on the same graph as your experimental data.