Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The ring of trigonometric functions over $\mathbb{R}$ is the ring generated by $\sin{x}$ and $\cos{x}$.

What's the reason for why any function $f$ in this ring can be written as $$ f(x)=a_0+\sum_{k=1}^n(a_k\cos{kx}+b_k\sin{kx}) $$ for $a_0,a_k,b_k\in\mathbb{R}$?

share|cite|improve this question

1 Answer 1

Observe that $\mathbb{R}[\sin x, \cos x] \subset \mathbb{C}[e^{ix}].$ Hence for any element $f \in \mathbb{R}[\sin x, \cos x]$ there exist elements $a_i\in\mathbb{C}$

$$f(x) = \sum_{i=0}^{n} a_n(e^{ix})^n = \sum_{i=0}^{n} a_n(\cos nx + i \sin nx).$$

Equating the real parts of both sides we obtain

$$f(x) = \sum_{i=0}^n \operatorname{Re}(a_n)\cos nx + \operatorname{Re}(ia_n) \sin nx,$$

as desired.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.