Differential Equations: Laplace Transform of Sines and Cosines
The Laplace transform of sin(at) and cos(at) is typically presented as a particularly unintuitive repetition of integration by parts. Sine and cosine, however, with the aid of Euler's formula, can be rewritten as a scaled sum of imaginary powers of e. Knowing the Laplace transform of e raised to any power, the Laplace transform of sin(at) and cos(at) becomes clear.