I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one might use these to solve some sort of differential equation faster/easier than other methods.
This does not have to relate to differential equation examples either. I'm just looking for some examples(preferably real world!) where one might want to take the sine or cosine of a matrix. Any ideas?
Thanks!
In Quantum Mechanics (when working with spin) one often encounter expressions like $e^{ian_i\sigma_i}$ where $\sigma_i$ are $2\times 2$ complex matrises (Pauli Matrices). This describes a rotation of the spin over an angle $a$. This expression is given by the $\sin$ and $\cos$ of a matrix since
$$e^{ia\vec{n}\cdot\vec{\sigma}} = \cos(a\vec{n}\cdot\vec{\sigma}) + i\sin(a\vec{n}\cdot\vec{\sigma})$$