trigonometry - Functional equation involving sine function

Let $f: \Bbb R \to \Bbb R$ be a continuous function such that $\sin x + f(x)= \sqrt 2 f\left( x- \frac {\pi} {4} \right) $. Find $f$.

I noticed that a solution for $f$ is the cosine function. I don't know how to continue. Is there a way I could link it to D'Alembert functional equation?