You mentioned phase constant, angular frequency and time. So, it must be a wave function. But I did not understand why you mentioned cos function since wave is a sine function.
To understand waves, we should understand simple harmonic motion(SHM) first. It is a type of to and fro periodic motion. Velocity and thereby kinetic energy of object undergoing SHM is nil at two extreme positions and maximum at mean position. Potential energy is maximum at extreme positions and nil at mean position. So, it can be thought of as conversion of two forms of energy in pace of time. Motion of pendulum is a example of SHM. The motion is so uniform (continuous in mathematics) that it follows precise mathematical model i.e. sine function.
Projection of uniform circular motion along it's diameter was studied by mathematicians and it was obviously a sin function. Amazingly, it exactly matched with motion of simple pendulum. You can set simple pendulum into motion, note it's period then you can calculate displacement from mean position at any point of time fitting values in 'motion of diameter projection of circle' formula.
Due to similarity of diameter projection, to and fro motion of simple pendulum is sin function.
Now, come to waves part. Waves transfer energy by oscillation of medium. Medium oscillates exactly like simple pendulum. Wave function is composed of two parts. Amplitude function and phase function multiplied together. First is amplitude part. It is constant, so no need to consider as function of any variable. Maximum displacement(extreme point)from mean position is called amplitude (A) which is multiplied ahead of sine function. Second is phase part. Phase ranges from -1 to +1. You can make analogy of phase part with unit vector which provides direction with magnitude of 1. Omegatime means angular velocitytime. Angular velocity remains constant. So, the only variable is time.
One important point to understand is you can not directly compare SHM of particle with angular velocity and time.(concept of angular velocity applies only in circular motion.)So,first reverse project SHM of wave particle into circular motion as we did in first part of answer, then you will be able to relate SHM with angular velocity *time.
I explain phase difference now. Suppose two particles represent two separate parallel sound waves. One is in mean position and other is in extreme positions, there will be 180 degree phase difference if we project it into circle. Their energy (Kinetic energy of first and potential energy of second) get subtracted. Conversely if both are at mean position or both at extreme position, energy gets added. You can imagine in between the extremes.
Note that omega*time is angle and fi is also angle. So two angles can be added.