Angular Frequency (ω = 2πf)

One cycle is $2\pi$ radians. Frequency counts cycles per second; angular frequency counts how fast the phase advances, in radians per second. Because one full cycle carries the phase through $2\pi$ radians, $\omega = 2\pi f$, and the two quantities differ by that factor alone. It is the $\omega$ that sits inside $\sin(\omega t)$, in wave and circuit equations and in the harmonic constants of a tide table, so a formula written in $\omega$ takes this conversion as its first step.

Reading the answer. The number in radians per second is larger than the number in hertz by the factor $2\pi$, and mains frequency shows it: 60 Hz is 377 rad/s and 50 Hz is 314 rad/s, the textbook radian values. Run backward, $f = \omega/2\pi$ returns 60 Hz from 377 rad/s. The page solves for either quantity, so you can start from whichever one you have.

A tide table in awkward units. The IHO and NOAA tidal constituent tables give the constituent M2 a period of 12.4206012 h, which is a frequency of 0.0805114007 per hour, and a speed of 28.9841042 degrees per hour, the same angular frequency written in degrees. To reach radians per second, multiply the degrees by $\pi/180$ and divide by 3600, which gives 1.4051890e-4 rad/s. Entering the frequency of 0.0805114007 per hour and asking for $\omega$ gives that same 1.405189e-4 rad/s, with the page handling the hours for you.

Limits. The relation is exact: it follows from one cycle being $2\pi$ radians, so there is no approximation to tolerate and nothing to calibrate. The usual slip is the unit, not the arithmetic, since a frequency in hertz and an angular frequency in radians per second are different numbers for the same oscillation. Related pages give $\omega$ from the period as $2\pi/T$, link the period to the frequency as a reciprocal pair, and use $\omega$ for resonance in an LC circuit.