Method for determining probe angle by analyzing skewed sine wave
I will assume:


The probe is like a line, infinitely narrow.
We move the probe along the $x$ dimension, parallel to the large-scale surface, and from each position, probe the surface:
The probe is extended to the direction it is pointing to until it meets the surface. (An alternative assumption would be that probing is done to the direction $y$, that of the normal of the large-scale surface, in which case the waveform measured would be sinusoidal unless the probe stem meets the surface before the tip. I wonder if your edit describes this scenario.)


Equation of line starting at coordinates $(x_0, 0)$ and with a positive slope $k = \frac{\Delta y}{\Delta x} \ge 1$ (where $1$ is the maximum slope of the sinusoid of Eq. 2, to ensure continuity):

$$y = k (x - x_0)\tag1\\
\Rightarrow x = x_0 + \frac yk.$$

Equation of the first period of the constant-biased sinusoid, spanning $0 \le x \le 2\pi$:

$$y = 1 - \cos(x)\tag2\\
\Rightarrow x = \begin{cases}
\arccos(1 - y)&amp;\text{if }0 \le x \le \pi\\
2\pi - \arccos(1 - y)&amp;\text{if }\pi \le x \le 2\pi.
\end{cases}$$

By Eq. 2 the period starts at $(0, 0)$ and ends at $(2\pi, 0)$, which makes the corresponding period of the skewed sinusoid also start and end at these points. We obtain the waveform by solving $y$ as function of $x_0$ from the system of Eqs. 1 and 2. Unfortunately there is no closed form solution, but we can still make some plots:

Figure 1. The skewed waveform for different values of the constant $k$. At $k=\infty$, the waveform is equal to the constant-biased sinusoid. The waveform shown represents the probe tip $y$ position as function of the coordinate $x_0$ at which the probe shaft crosses the $y$ axis. If you desire the "probe length" waveform, you need to multiply this waveform by a factor $\tfrac{\sqrt{k^2 + 1}}{k}$.

I doubt that you will be able to find an analytical solution in frequency domain. What we can do is determine $k$ based on the waveform peak location $x_0 = x_p$ by setting $y = 2$, for which the system of Eqs. 1 and 2 gives:

$$k = \frac{2}{\pi - x_p}.\tag{3}$$

To get the angle $\phi$ by which the probe angle deviates (moderately) from the normal of the large-scale surface:

$$\phi = \operatorname{atan2}\left(1, k\right) = \operatorname{atan2}\left(\pi - x_p, 2\right).\tag{4}$$