The exponential function$\exp: \mathbb{C} \to \mathbb{C}$ relates to sine and cosine$\cos, \sin \colon \mathbb{R} \to \mathbb{R}$ via

$\exp(i x) = \cos(x) + i \sin(x)$

This can be interpreted as either a theorem (especially when $x$ is restricted to real numbers and $\cos, \sin$ have been defined independently of $\exp$), or as an implicit definition of $\cos, \sin$ (where more explicitly we have $\cos(x) \coloneqq \frac1{2}(\exp(i x) + \exp(-i x))$ and $\sin(x) \coloneqq \frac1{2 i}(\exp(i x) - \exp(-i x))$).