How to tell if an equation is a wave?

Good question.

If the equation has a factor in it that represents up/down or back/forth oscillation, then it's a wave function. Note that the oscillation might be periodic or aperiodic, makes no difference. It's still a wave.

So if the equation has, say, a frequency over time factor, that's a wave equation. Frequency over time implies repetition, which is to say, oscillation. And as, by definition, the period is the time to complete one full oscillation, one cycle, an equation with a period factor in it is also a wave equation.

Trig functions with angular speed W = 2piF and time t are great examples of wave functions. For example, a = A sin(Wt) = A sin(2piF t) = A sin(2pi t/T) where F = 1/T is the frequency and T is the wave's period. If you plot this out from t = 0 to t = T, the plot will be one wavelength of, ta da, a sine wave. In this case, the oscillation is up down with A as the max up and max down, A is called the amplitude of the sine wave.

NOTE: Kyle's answer is incomplete. There are waves other than sine and cosine waves. For example, so-called square waves are generated to power radars. And saw-tooth waves are not uncommon in EE. And these waves are not...not...defined by either of the two trig functions.

it will either have Sin( _ ) or Cos ( _ ) in it.

It could also have the inverses of those as well.

There must also be a theta sign inside the parentheses.