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Is there a way to compute the coefficients $a_k$ in series of the form $\sum\limits_{k=1}^{\infty}a_k \sin(ka) = b$ and $\sum\limits_{k=1}^{\infty}a_k \sin(ka) k = b$ for fixed constants $a,b\in\mathbb{R}$? I thought about Fourier series of course but without dependence of $x$ of the sin term I am stuck... Thanks.

Sebastiano
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HelloEveryone
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    Could be as simple as assuming $x=1$ in the standard Fourier series form (or Fourier sine or Fourier cosine series, if you prefer), i.e. derive a general representation for a certain function $f$, and then let $x=1$. – PrincessEev Nov 29 '22 at 08:40
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    There is not a unique way of doing that. You can simply pick $a_1=b/\sin(a)$ and $a_k=0$ for $k>1$. Note that in the exceptional case when $a$ is a multiple of $\pi$, then the series is identically zero and there is no solution if $b\neq 0$. – Lorenzo Pompili Nov 29 '22 at 08:48

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