‎In this paper‎, ‎we present some characterizations of linear mappings‎, ‎which preserve vectors at a specific angle‎. ‎We introduce the concept of $-varepsilon‎, ‎c-$-angle preserving mappings for $|c|<1$ and $0leq varepsilon < 1‎ + ‎|c|$‎. ‎In addition‎, ‎we define $widehat{varepsilon},-T‎, ‎c-$ as the ``smallest\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\'\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\' number $varepsilon$ for which $T$ is $-varepsilon‎, ‎c-$-angle preserving mapping‎. ‎We state some properties of the function $widehat{varepsilon},-.‎, ‎c-$‎, ‎and then propose an exact formula for $widehat{varepsilon},-T‎, ‎c-$ in terms of the norm $|T|$ and the minimum modulus $[T]$ of $T$‎. ‎Finally‎, ‎we characterize the approximately angle preserving mappings‎.

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