What is the meaning of $x(t)=x_0\cos(\omega t+f)$, where $x_0$ is the amplitude, $\omega$ the angular frequency, $t$ time and $f$ phase constant?
I know how to solve the mathematical problems which use this expression, but what I don't understand, how $\sin$ and $\cos$ functions are used to express $AC$ waves. I know the math but I do not understand the theory behind it. I do not have an intuitive understanding of these expressions. If you cannot explain, at least guide me to a resource that can.
I want to understand the math in a theoretical manner like are we going to take the $\sin$ / $\cos$ of the $\omega t$ or the solved value of it, since $\sin$ / $\cos$ is a ratio so basically are we multiplying the $\sin$ / $\cos$ fraction value to the variables, oh i am so confused please explain .
Let's start from the beginning. I'll throw some plots together in Mathematica, and hopefully this will help.
To start with, you have your basic sine wave, $\sin ( x )$, and your basic cosine wave, $\cos ( x )$:
Then you can multiply it by something on the outside, like in $5 \sin ( x )$ or $.1 \cos (x)$, and the vertical dimension is multiplied by that factor, because this just multiplies the entire sine or cosine by that factor:
Note that here the scale has changed, according to the factor out front.
You can also multiply by a factor inside of the function, like in $\sin (5 x)$ or $\cos(0.5 x)$, which $\textit{divides}$ the horizontal dimension because you are now changing the input to the sine or cosine. $\sin ( 2x)$ should change twice as quickly as $\sin ( x )$, because when $x$ goes from $a$ to $b$, $2x$ goes from $2a$ to $2b$, which is twice as much of a change:
These two are on the same scale as the original; note that they are still ranging from $-1$ to $1$, because we didn't use a multiplicative factor out front.
The next part is when a constant is added inside of the function's argument, like in $\sin (x + \pi)$ or $\cos (x - \frac 12 \pi)$ or $\cos (x + \frac 13 \pi)$. Here you are "shifting" the argument, hence the phrase "phase shift", and this is why the constant added inside is sometimes called the phase of the wave. The only result of this change is that the entire curve moves to the side. If you replace $x$ with $x + c$, then at every point $x$ on your new curve, you get the same value as the point $x + c$ on your old curve.
Note that adding or subtracting $\pi$ moves you half of a period down the $x$-axis, so that $\sin (x + \pi) = \sin(x - \pi) = - \sin(x)$. Also, $\sin (x)$ and $\cos(x)$ are the same shape, just shifted by $\frac{\pi}{2}$, so that $\sin(x - \frac{\pi}{2}) = \cos (x)$ and $\cos(x - \frac{\pi}{2}) = \sin (x)$.
Once you have all these pieces, you can throw them all in the same function. For example, $3 \sin (2 x + 1)$ takes $\sin (x)$, stretches it vertically to range from $-3$ to 3, rather than $-1$ to 1, doubles the period (compresses it horizontally by a factor of two, because it changes twice as quickly now), and shifts it to the left 1. Thus:
All of these changes can be applied just as easily in terms of $\omega, x_0$, and $f$.
For an AC wave, you have a source, which outputs a voltage to the circuit. Since this is AC, the voltage is not constant in time. Usually the voltage is represented by a sine or cosine wave. Thus your voltage from the source is of the form $x_0 \cos(\omega t + f)$. This means that the minimum and maximum voltages are $\pm x_0$, the angular frequency (radians per second) is $\frac{1}{\omega}$ (making the frequency $\frac{1}{2 \pi \omega}$, which is in Hz), and the phase is $f$. Some circuits change the form of the wave before the signal is output. Some circuits amplify the wave, changing $x_0$. Some circuits delay the wave, changing $f$. Some can differentiate it, and some can integrate it, changing the shape of the wave entirely. Others might squish some parts more than others, and there are circuits that can chop off the tops or bottoms of the waves, leaving a flat segment. The more complicated the circuit, the more ways it can change the wave.
Usually more convenient than sines and cosines is the complex exponential. We can use Euler's Formula, $e^{ix} = \cos(x) + i \sin (x)$, to express these waves as the real part of an exponential: $x_0 \cos (\omega t + f) = x_0 \Re (e^{i(\omega t + f)})$, where $\Re$ means to take the real part. This is much easier to manipulate when working with equations. To recover the original wave, simply take the real part, and there it is.
The sines and the cosines are just ways to mathematically handle what the AC current does: it waves. When something waves, you model it with a wave. The simplest waves are sines, so we start with those. Thanks to Fourier we can stick to sines, and all other waves are just linear combinations of these sines. Thus if you want to model AC circuits, you need to model waves, so you use sines and cosines.