Does 1 kHz sine tone means $\sin(2(1000)\pi t)$ or $\sin(2(500)\pi t)$?
The trigonometric functions "do not know" what a Hertz is and they do not care either. The only thing they know is that a full circle is $2 \pi$ radians. Whether this circle concludes in days, hours, picoseconds or a slice of it represents the angle a force is applied to some lever, is immaterial.

$2 \pi \omega$ expressed in Hertz, denotes a rate. A rate of going around a circle at the time span of a second. $y = \cos(2 \pi 1 t)$ where $t$ is in seconds, would have concluded 1 circle, composed of $2 \pi$ radians, by the time $t$ ticks to 1.

To make it conclude the circle faster, we multiply the "passing of time" (denoted by $t$) by some number $f$.

Therefore, a 1kHz tone is $2 \pi 1000$ radians per second.

Hope this helps.