Given $r \in \mathbb{R}$ and $k \in \mathbb{N}$, it is said in an article that the Fourier series of the function $\exp\big(ir \sin(kt)\big)$ is $$ \sum\limits_{m=-\infty}^\infty J_m(r) e^{imkt},$$ where $J_m(r)$ is the Bessel integral $$ J_m(r) := \frac{1}{2\pi} \int_0^{2\pi} e^{ir \sin(\theta)}e^{-im\theta}\; d\theta.$$
If I'm not mistaken this means that the $mk$-th Fourier coefficient ($m \in \mathbb{Z}$) of the function $\exp\big(ir \sin(kt)\big)$ is $J_m(r)$ and the other Fourier coefficients (those that are not multiples of $k$) are zero.
Using a change of variable, I could show that the $mk$-th Fourier coefficient of the function $\exp\big(ir \sin(kt)\big)$ is $J_m(r)$. However I'm unable to show that the others are zero.
How do we show that the following integral is zero for all $m \not\in k \mathbb{Z}$ ?
$$\frac{1}{2\pi} \int_0^{2\pi} e^{ir \sin(kt)}e^{-imt}\; dt.$$