what is angular frequency ! what does it mean?

I know that

$$\omega = 2 \pi f$$

why do we use angular freq to expression about the instantaneous values of the wave?

• The instantaneous values of the wave are governed by the phase, not the frequency. – Yves Daoust Apr 23 '18 at 20:19
• Also have a look at this question. – Gilles Apr 23 '18 at 20:58
• What angular frequency means in simple words . I can define the simple frequency by number of cycles per second I am asking about the physical meaning ! –  jemy Apr 23 '18 at 22:09
• Did you read my answer? Do you know what radians are? Between this and your other question, it seems that you need to understand the sine and cosine functions a little bit better as a start. – Cedron Dawg Apr 23 '18 at 22:22
• So could you recommend any good materials to me I will be so thankful for your help –  jemy Apr 23 '18 at 22:29

It's easiest to understand with units. The argument for the sine and cosine functions are in radians. Thus angular velocity represents the frequency in the most convenient way for these functions. For instance, if your time unit is seconds, then the units for $f$ are cycles per second, aka Hz. The $2\pi$ has units of radians per cycle, so the equation you mentioned $\omega = 2 \pi f$ has units of:

$$\frac{radians}{second} = \frac{radians}{cycle} \cdot \frac{cycles}{second}$$

When it is plugged into the equation for your signal:

$$s(t) = A \cos( \omega t + \phi )$$

The argument $\omega t + \phi$ has units of:

$$\frac{radians}{second} \cdot seconds + radians = radians$$

Hope this helps.

Ced

• radians as "units" is maybe not the best expression. – robert bristow-johnson Apr 24 '18 at 2:53
• @robertbristow-johnson, I'm baffled by you saying that. A radian is a unit distance along the circumference of a unit circle on the same scale as the underlying Cartesian coordinates. It can also be used as an angle measurement for the angle that subtends that arc. Why is "unit" not a very good description of that? – Cedron Dawg Apr 24 '18 at 3:17
• angles expressed (or measured) in radians are dimensionless numbers. pure numbers. meaning no units attached. for a circle of radius $r$, when an arc of length $s$ is swept, $\theta$ is simply the ratio, the dimensionless constant of proportionality relating $s$ and $r$: $$s = r \theta$$ $s$ is in units of length, $r$ is in unis of length, and $\theta$ is simply the dimensionless number that scales $r$ to equal $s$. degree is not really unit, but simply a dimensionless constant. attaching the degree symbol "°" is simply multiplying by the pure number $\frac{\pi}{180}$. – robert bristow-johnson Apr 24 '18 at 8:08
• nepers are not units either and neither are decibels. the dB "unit" has the same kind of role that the degree symbol ° has. attaching "dB" to a quantity is simply multiplying that quantity by $\frac{\log(10)}{20}$.  likewise percent is not a unit. attaching the "%" symbol is the same as scaling by the dimensionless constant $\frac{1}{100}$. that's all it is. – robert bristow-johnson Apr 24 '18 at 8:13
• @robertbristow-johnson, Thanks for all the replies. I've been down this path before. Units aren't the same as dimensions. I refute you thus: "The International System of Units assigns special names to 22 derived units, which includes two dimensionless derived units, the radian (rad) and the steradian (sr)." from en.wikipedia.org/wiki/SI_derived_unit. Conversion factors, which convert from one unit to a different unit in the same dimension, are by definition dimensionless. A cycle is dimensionless and a radian is dimensionless, yet 2*Pi is still the conversion factor. – Cedron Dawg Apr 24 '18 at 11:48