In Exercise 5.17 we saw how combinations of trig functions could be used to describe oscillations that occurred in layers, with a faster oscillation superimposed on a slower one. In this question we explore another way in which trig functions can be used to model complex oscillatory behaviour.

NOAA, the National Oceanic and Atmospheric Administration, is a US government agency that monitors data on the health of the oceans and the atmosphere. For example, they measure the amount of carbon dioxide, CO2, in the atmosphere. Typical data, taken from their web page in August, 2021, are shown in the upper panel of Fig. 5.23. Our goal is to find a combination of trig functions that can describe these data.

a. First, we need to remove the underlying trend. Calculate the equation of the linear trend shown by the dashed black line in Fig. 5.23. Keep this function, as you’ll be using it later.

b. If you subtract the linear trend from the data you get the modified data shown in the lower panel of Fig. 5.23. Find a function of the form Asin(ωt + φ) that describes this data approximately.

c. Now, the simple sin function doesn’t do a great job of describing these data, but if we take a more complex combination of sin and cos functions we can get a much better approximation. You don’t know how to do this, so you just This is actually a simple example of a large and important area of scientific analysis and applied mathematics, called Fourier series. This is a technique that is taught in more advanced math courses, and is a way in which complex oscillatory functions can be expressed as combinations of sin and cos.
need to trust us when we say that one possibility is the function [CO2] = a0 + a1 sin(ωt) + b1 cos(ωt)
+ a2 sin(2ωt) + b2 cos(2ωt), where a0 = −0.13252, a1 = 2.1446, a2 = −0.70382, b1 = 0.64646, b2 = 0.31341, and ω = 6.2801.
i. What are the units of these constants?
ii. Plot this function on the same graph as the sin function you obtained in part b above. You should get something like that shown in the upper panel of Fig. 5.24.

d. Finally, add your trend line (that you calculated in part

a) to the two fits and plot your two approximations of the data.
You should get something that looks like the lower panel of Fig. 5.24.