I have been asked to collect tidal data (time and height) for a location over a seven day period to answer the following question...
"Model the scenario using a single trigonometric function.
- Plot the height verse elapsed time in a suitable graphing software and interpret the observable amplitude, vertical shift and period of the data set, to fit a trigonometric model to the data. Provide reasons for your choices."
I'm assuming they want a sinusoidal function that best fits the data since the minima and maxima, over the 7 day interval, vary quite a bit and cannot be accurately modelled with a single trig function.
However, the question later asks to predict heights of high and low tides at a future time, which will be difficult given the inaccuracy of the model.
Any advice will be much appreciated.
You have a regression task.
We can't and we are not supposed to be able to through all the points due to the errors that you mentioned earlier. However, you can aim to minimize some sort of objective function.
If $(x_i, y_i)$ are your data points, then you can chosoe $A, B, C, D$ to minimize
$$\sum_{i=1}^n\left( y_i - A-B\sin (Cx_i +D) \right)^2.$$
If you can fit the curve perfectly, the quantity will be positive but we just want the quantity to be as small as possible.
Furthermore, you might also want to get rid of outliers.