# Using Trigonometry to Estimate Influenza Deaths

In a nutshell, the sin() and cos() terms are periodic curves, and the weighting of the various coefficients is what allows proper regression fits.  Let’s take a closer look at the formula, and try to make sense of it.

As t (weeks) increases to 52, $$\frac{t}{52}$$ goes from 0 to 1  ($$\frac{0}{52}$$, $$\frac{1}{52}$$, $$\frac{2}{52}$$, $$\frac{3}{52}$$, …, $$\frac{52}{52}$$)  Once it goes past 52, it just cycles around again.  Recall $$2\pi$$ radians = 360 degrees.  Since  $$\frac{t}{52}$$ is multiplied by $$2\pi$$, it is multiplying 360 by some number.   So, it seems the sin() and cos() terms simply use t weeks to scale across multiples of 360 degrees .

For example, as t goes from 0 to 52, $$\frac{t}{52}$$ goes from 0 to 1, $$2\pi * \frac{t}{52}$$ goes from 0 and 360.  (and then it repeats since sin repeats in multiples of $$2\pi$$ and therefore $$sin(2\pi * \frac{t}{52})$$ goes from sin(0) to sin(360) which is a full periodic cycle of this function.  Note the same logic applies to the cos() term in the formula.

The picture says it all.