Fourier series
In the 18th century, mathematicians such as Euler, Lagrange and Bernoulli already used sinusoids to approximate and model other functions. When Fourier published a work on heat, in 1822, he said that such approximations exist for any such function (that is continuous in the interval). At first, people didn't believe him, and it took almost ten years for a proof (of part of the problem) to appear.