= Narrowband frequency modulation
When the <FM modulation index> satisfies $|\beta|\ll1$, a <Taylor expansion> gives a useful first-order approximation. With $\omega_c=2\pi f_c$ and $\omega_m=2\pi f_m$,
$$
\frac{s(t)}A
=\cos(\omega_ct)-\beta\sin(\omega_mt)\sin(\omega_ct)+O(\beta^2)
=\cos(\omega_ct)+\frac\beta2\cos((\omega_c+\omega_m)t)-\frac\beta2\cos((\omega_c-\omega_m)t)+O(\beta^2).
$$
The <product-to-sum formula> for <sines> exposes a carrier and one pair of sidebands, with opposite signs for this <wave phase> convention. Higher sidebands are neglected, not identically absent. For $\beta=0.1$, the predicted normalized first-sideband magnitude $0.05$ is close to the original exact evaluation $J_1(0.1)\approx0.0499375$. The neglected waveform term is bounded in <absolute value> by $\beta^2/2=0.005$, using the second-<derivative> bound for the <cosine>. Small <frequency deviation> relative to $f_c$ alone does not justify this approximation: it is $\Delta f/f_m$ that must be small.
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