I know that
$$ \omega = 2 \pi f $$
why do we use angular freq to expression about the instantaneous values of the wave?
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$$ \omega = 2 \pi f $$
why do we use angular freq to expression about the instantaneous values of the wave?
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