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**Isomorphism** Oh, I answered that in a general sense, because you talked about digitalisation of sin and cosine. If you want practical applications there, the entire branch of Digital Signal processing (which does audio processing, video processing etc) deals with Laplace transform and design of filters using it.

However as TPH explained, it is a cunning way to solve differential equations

(if you find algebra easy).

However I would like to add one point. For engineers(or physicist) every ODE has an initial conditions specified. Laplace takes care of that too. It is thus a good enough reason to learn Laplace transforms and apply them.