Let \(\xi \in \mathbb {R}\) be an irrational number and \(\eta \in \mathbb {R}\) . Dirichlet’s theorem on Diophantine approximation and its inhomogeneous version, due to Kronecker, tells us that \(|\xi -m/n|=\mathcal {O}(1/n^2)\) and \(|n\xi -m-\eta |=\mathcal {O}(1/n)\) for infinitely many pairs of integers \(n,m\in \mathbb {Z}\) . We derive two refinements telling us that for any \(A>0\) , \(\rho >0\) , \(\eta \) rational, and \(0<\varepsilon <2\) there are infinitely many pairs \((n,p)\in \mathbb {N}^*\times \mathbb {Z}\) such that \(\begin{aligned} n^{1-\varepsilon } |n\xi -p-\eta | \sim A, \quad \text{ resp. } \quad n^{2(1-\varepsilon )}\left( n\xi -p-\eta \right) ^2 \sim \rho \log n+A. \end{aligned}\) Then we apply these facts to revisit results by F. Luca and J.C. Saunders on the set of cluster points of sequences of the form \(f^n(\sin (\alpha n))\) and \(n^{s} |\sin (\alpha n)|^{n^{r}}\) and their analogues for the cosine function. A very special case e.g. will tell us that for every \(\alpha \) for which \(\alpha /\pi \) is irrational, the cluster set of \(\begin{aligned} \big (1-\sin ^6 (\alpha n)\big )^{n^5} \end{aligned}\) is [0, 1], whereas this no longer holds for \(\begin{aligned} \big (1-\sin ^6(\pi \sqrt{2} n)\big )^{n^6}. \end{aligned}\) Several results depend heavily on the irrationality exponent associated with \(\alpha /\pi \) . In the last section we study the asymptotic behavior of \(|\sin (n_k+m_k)|^ {\varepsilon _k}\) for various sequences \(\varepsilon _k>0\) whenever \((\sin n_k)^{n_k}\) and \((\sin m_k)^{n_k}\) converge to non-zero numbers. It will finally be shown that for \(0<|\lambda |<1\) the existence of the limit \(\lambda :=(\sin n_k)^{n_k}\) implies that \(\lim _k \big |\sin (p n_k)\big |^{pn_k}\) exists, too, and equals \(|\lambda |^{p^3}\) if p is odd and 0 if p is even.
