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For different values of $\gamma \geq 0$, analysis of the end behavior of the sequence $a_n = \cos (n)^{n^\gamma}$ yields a strong connection to the irrationality measure of $\pi$. We show that if $\limsup |\cos n|^{n^2} \neq 1$, then the irrationality measure of $\pi$ is exactly 2. We also give some numerical evidence to support the conjecture that $\mu(\pi)=2$, based on the appearance of some startling subsequences of $\cos(n)^n$.

The irrationality measure of $\pi$ as seen through the eyes of cos($n$)