Regarding the equation derivations of the resonant state of an asynchronous buck converter observing zero voltage switching

Question

I'm currently reading this article of Bill Andreycak to understand the mathematics behind zero voltage switching: https://www.ti.com/lit/an/slua159/slua159.pdf?ts=1710666811170&ref_url=https%253A%252F%252Fwww.google.com%252F. At present, I'm currently reading on the ZVS Design Equations portion.

When it comes to the derivation the design equations for the initial conditions and the capacitor charging state, I feel like I totally got the hang of it. However, I am not sure I totally understand the derivations/circuit analysis for the resonant state.

First of all, here's my understanding of the resonant state:

At t₁

1. This state is triggered when/starts after the resonant capacitor has has fully charged at t₁. At this very moment, i_Cr should be equal to I_OUT but it should be gearing to decrease to zero.
2. As the parallel combination of C_R and Q₁ would now theoretically behave like an open circuit as C_R have now been fully charged, the cathode of the freewheeling diode D₀ now effectively has a lower voltage received compared to its anode. In short, the freewheeling diode D₀ should now start conducting, triggering the output inductor to act as a current source (in other words, the voltage polarity of output inductor should flip).
3. The resonant inductor L_r also acts a current source (in other words, the voltage polarity of resonant inductor should flip) since the open circuit behavior of C_R and Q₁ at the beginning of this state would indicate a current interruption, although, of course, the direction of i_Lr should remain as is.
4. The resonant inductor current i_LR should be equal to IOUT at this specific period, but should also gear to decrease afterwards. It is also equal to i_CR.

Between t₁ to t₂

1. The resonant capacitor current i_CR and the resonant inductor current i_Lr continues to be equal, however, they're no longer equal to the output current I_OUT. As mentioned, i_CR and i_Lr is expected to be decreasing throughout this duration (or at least it's how I understood it).
2. The resonant inductor voltage V_LR should also be decreasing to some value throughout this duration.

With that understanding, this is how I would implement polarities and directions in the schematic diagram at resonant state for circuit analysis:

Now, when I try to derive the equations myself for the t₁ to t₂ duration, although it feels like I got the KVL right, I don't think I understand the decision behind designating the sine function to the resonant inductor voltage and the cosine function to the resonant inductor current. I mean, ELI the ICE man right? So following that, the resonant inductor voltage should follow a cosine function while the resonant inductor current should follow a sine function, right?

If you check out the article, you would see that these are the derived equations during resonant state:

I mean they do mention that the current should follow a cosine function but proceeds to not give an explanation why:

If somebody could explain to why the design equations for the resonant state are the way they are, it would greatly help me. Perhaps somebody could generously go over the proper way of circuit analysis for me because that maybe the reason why there's a disconnect from my end.