What are some examples that require the use of Sinusoidal Regression? I am interested in knowing where do regression models of the form:
$$
\phi(x,A,B,C)=A\sin(kx+B)+C
$$
for a given $k\in\mathbb{R}$ are applied in domains such as engineering and science. What example are there that requires the use of this model?
I had encountered this model recently on math stack exchange. I would suspect the use of this model would be in a domain that heavily requires the use of signal processing but I can't seem to find any answer on the web of any application of sine regression.
These questions on MATHSE are mainly about how to transform the nonlinear sine model to a linear one via substitution in order to be solved numerically (such as using the normal system)
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
https://math.stackexchange.com/questions/2368821/find-a-sinusoidal-regression-equation-for-some-given-data

We have a very sensitive optics system¹, where our signal changes slowly over time. However, there is a background noise of 60hz caused by nearby AC current. We sample 25 points over a second, and recover data like:
time  signal
0.011 1546
0.044 1615
0.081 1772
0.115 1795
...
0.802 1792
0.838 1498
0.873 1507


Plotted, it looks like:

Note that the periodic behaviour above is aliasing from the 60hz signal.
We could just take the sample average of all points to get a estimate of the signal, but this isn't efficient: consider what happens when the samples start and end both “high”, or both “low” - this biases the average up, or down, respectively.
Instead, we directly model the system using the sinusoidal regression:
$$f(t, A, B, C) = C + A \sin(120\pi t + B) $$
Like the links in the question, we expand this as:
$$f(t, A', B', C) = C + A' \sin(120\pi t) + B'\cos(120\pi t) $$
which turns our problem into a linear regression problem. Solving using least squares and converting back to the original parameterization gives:
C = 1676.40
B = 1.46
A = 200.74

Plotting the fitted model vs the observed:

We use the value of C, 1676.4, as our final estimate. Note that this is different from the sample average of 1654.0.
¹ We are measuring low light signals that bounce of cells in our bioreactors.
