# Voltage Equation in Time Domain

When we describe a AC signal (voltage is this case) in time domain we use an equation something like this below:

$$\v(t) = V_m \mathrm{sin}(\omega 𝑡 + \phi)\$$, where 𝜔 is the angular frequency of the voltage.

I know 𝜔 is the symbol for angular frequency which is equal to $$\2 \pi f\$$ and is measured in rad/s.

So why does the voltage equation above show the angular frequency as 𝜔𝑡 and not just 𝜔? Is the $$\t\$$ there just to show the measurement is with respect to time?

Or is it because 𝜔 and 𝜔𝑡 not the same?

No, $$\\omega\$$ and $$\\omega t\$$ and are not the same, and they don't have the same units. $$\\omega t\$$ has units of radians, which makes it possible to add the phase angle $$\\phi\$$ to it and get something sensible.
In the equation you provide, the values of $$\V_m\$$, $$\\omega\$$, and $$\\phi\$$ are constants. So, if we want to find the voltage as a function of time, then there must be a variable in the equation that represents the specific value of time in question. That variable is $$\t\$$.