[Year 12 Trigonometric Functions] Research assignment investigation, using tidal waves to model data. 

The behavior of tidal waves is more complex than what can be represented by a simple cosine or sine function because many factors influence them. Tidal patterns can vary greatly and do not always follow a simple periodic function.

For part (b) of your question, the use of the function

$y=ksin(at+θ)+lsin(bt+θ)+c$

Can better model the data for several reasons:

It allows for the superposition of waves with different amplitudes, frequencies, and phases, which allows simulating the overlapping effects of different tidal forces.

Using different amplitudes (k and l) can represent the varying strength of tidal forces at different times.

Also, different frequencies (a and b) can account for the fact that tides are influenced by multiple factors

More importantly, the phase shifts (θ) allow for the representation of the delay.

Adding a constant (c) can shift the entire function up or down to match the mean sea level.

So, this formula allows for the modeling of two different tidal forces acting with different intensities and periodicities, thus giving a more accurate representation of the actual tidal behavior.

For part (c), you can approach modeling this data by:

Identifying key points in the data, such as the times and heights of high and low tides.

Then, you can use these points to estimate the amplitudes (k and l), frequencies (a and b), phase shifts (θ), and vertical shift (c).

Inputting those initial estimates into a graphing software (like Desmos) and adjusting them iteratively to fit the provided data as closely as possible, is the way to go.

For example, you can assign initial values to the constants $k,a,θ,l,b, and, c$, like starting with:

$k=l=1$, $a=b=2π​/T$

where T is the approximate period you observe in the tide data, and

$θ=c=0$ as a starting point.
 
You can then adjust the values of these constants to see how the graph changes. You can see how the graph stretches vertically with changes to k and l, shifts horizontally with changes to θ, and changes its frequency with changes to a and b. Remember that the process is iterative and requires some trial and error. You could also use software with fitting capabilities to automatically find the best values for these constants based on your tidal data.

In your graph/data, there are two high tides and two low tides shown, with the corresponding times indicated at the peak and trough of each wave. The first high tide peaks at around 1.5 meters just before 4 am, followed by a low tide at around 0.5 meters just before 11 am. The pattern repeats with the second high tide reaching nearly 2 meters just after 7 pm and the following low tide dropping to about 0.75 meters shortly before 10 pm. Also, there are two high tides and two low tides within 24 hours, so your main tidal frequency (associated with a or b) would be around 12 hours. And so forth...