Now we model seasonality. The amplitude is about the same as the slope, so we’d know just from that (even if we didn’t visualize it) that it’s going to make the line non-monotonic: there will be parts of the year when CO₂ concentrations are temporarily falling.
A sine wave with zero offset is rising at x=0, so the fact that this first phase term is close to zero tells us that CO₂ is on the upswing around new year’s, the zero of the calendar cycle. The source data is from the northern hemisphere, where plants are rotting most and growing least in midwinter. (The southern hemisphere’s seasonal signal is much smaller anyway because more of its photosynthesis happens in water instead of in air: https://en.wikipedia.org/wiki/Carbon_dioxide_in_Earth%27s_atmosphere#/media/File:Global_distribution_of_Carbon_Dioxide.jpg – Armstrong is at about 35° N.)
The seasonal cycle is not a pure sine wave, so we refine it with a frequency one octave higher. This gives the cycle a slight knee in the autumnal upswing and a straighter downhill shot in spring.
If you click the curve in Desmos, you get local extrema highlighted. It looks like the model puts the yearly peak CO₂ 37.3% of the way through the year, or mid-May, and the low at 71.7%, or mid- to late September.
