Frequency is usually given in Hertz: Hertz = 1/s
But using any units, yes it is 1/t.
You can think of it as "cycles/second" if you want (or any other time unit), but we just don't write units for cycles.
So yes it is for every case. the t is usually T which stands for the period, you just need to decide in every case what T is measuring.
Usually how long it takes for a wave to repeat itself.