First, the formulas by itself are from sine/cosine waves, so you should read a little about them first. Here.
For the rest, you only have to keep in mind the unities $\phi$ in $\frac{rad}{s}$, $\Delta$ in $rad$ and finally in x(n) is a discrete sine-wave (or sampled) that is why the $n$ in the formula. It is basically the same thing as with the $t$ but in discrete, or sampled, time.
The $T$ stands for the Period of the wave. You can perform simply math there to check the following:
$$
T = \frac{2k\pi}{\omega}
$$
$$
T = \frac{1}{f}
$$
$$
\omega = 2\pi f= \frac{2\pi}{T}
$$
Then you seen why we can obtain:
$$
T = \frac{2\pi}{\omega}
$$
Finally, the $k$ stands, normally, for each of the cycle the wave is repeated. Then you can have $k$ = 0, 1, 2, ..., N
To understand more how to work with this on DPS you only should read about Sine-waves and its representation. Basic knowledge in shifting and characteristics as Period, Frequency, Angular velocity and phase is a must.
Bests!