Finding the ideal phase shift when manually fitting a trigonometric curve

Question

So essentially for a school project I've been given a scatterplot with an array of points shaped in a rough sinusodial wave, and I need to plot a line of best fit manually.

So far I have all the values, except for c (horizontal/phase shift). I want to find the best way to get the most accurate value. so far my equation is this: y = 3.685 cos 29.1(x+c) + 5.455 Desmos approximates the value of c to 1.22, as can be seen here:

But obviously I have to calculate the phase shift myself so this isn't really helpful. Here is my graph so far: https://imgur.com/a/YCr9mjj

The curve in blue is desmos model, fitted to the points on the scatterplot. The one in red is what i've got, using the c value of 1.22 that desmos automatically calculates and plugs into my function.

How can I calculate the c value myself?

desmos file if anyone wants to play with this or see the values for themselves: https://www.desmos.com/calculator/us3bchvbwq

Attempts: I tried substituting x and y coordinates of a point on my graph (with phase shift set to 0), and then solving for c. My teacher showed me how to do it, apparently you can just stick a 360 in there, but im not sure how the math works. Here are the steps anyway:

This gave me a shift of around 27.1894, which resulted in an rho of 0.946 but I'm wondering if I can do better. The Desmos file is there with all the values, I understand the problem is complicated, so thank you in advance for anyone who takes time out of their day to help me!

Answer 1

Frame change:

Your data have only two or three significant digits. It makes no sense to look for or report model parameters with more decimal places.

So round what you have and proceed to interpret it in your project.

Answer 2

You need to fit the following curve

$$ y = A \sin( k x) + B \cos( k x) + C \tag{1}$$

for the parameters $(A,B,C)$.

Then the amplitude of the oscillation is $R = \sqrt{A^2+B^2}$ and the phase is $\phi = \arctan \frac{B}{A}$ since the above is equivalent to

$$ y =R \sin ( k x+\phi) +C \tag{2}$$