Which one of the following is the correct relation between frequency $

Which one of the following is the correct relation between frequency $f$ and angular frequency $\omega$?

$f = piomega$
$omega = 2pi f$
$f = 2omega/pi$
$f = 2piomega$
This question was previously asked in
UPSC NDA-1 – 2017
The question asks for the correct relation between frequency ($f$) and angular frequency ($\omega$).
Frequency ($f$) represents the number of cycles or revolutions per unit of time (usually per second), measured in Hertz (Hz).
Angular frequency ($\omega$) represents the rate of rotation or oscillation in terms of angle per unit of time (usually radians per second), measured in rad/s.
One complete cycle or revolution corresponds to an angle of $2\pi$ radians. If $f$ cycles occur per second, then the total angle covered per second is $f \times (2\pi)$ radians.
Therefore, the angular frequency $\omega$ is related to the frequency $f$ by the equation:
$\omega = 2\pi f$
Angular frequency is commonly used in physics and engineering, particularly when dealing with circular motion, oscillations (like simple harmonic motion), and waves. It simplifies many equations compared to using frequency directly, especially in the context of derivatives and integrals involving sinusoidal functions. The unit of angular frequency is radians per second (rad/s), whereas the unit of frequency is Hertz (Hz) or cycles per second.