Multiplying a signal by a sinusoidal oscillator is called mixing and translates spectral content to new frequencies.

For

$$y(t)=x(t)\cos(2\pi f_ct),$$

write the cosine as two complex exponentials. The Fourier transform becomes

$$Y(f)=\frac12X(f-f_c)+\frac12X(f+f_c).$$

The original spectrum is therefore copied around $+f_c$ and $-f_c$.

A transmitter can use this operation to move a baseband message into a passband. A receiver can mix the passband signal again with a synchronized oscillator; the product contains a component translated back toward zero frequency and another at a much higher frequency.

A low-pass filter can retain the baseband component and reject the high-frequency one.

Mixing is the common frequency-translation operation behind carrier modulation, coherent demodulation and many radio-frequency conversion stages.