# 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.