Question and a0

an, bn, f(x) and plot

Following the general equation as quoted from various reputable sources, such as John Bird, although the cos(x)/2 term makes perfect sense, I cannot understand its origin. I have proved it graphically to be the missing term, but mathematically I cannot.

Please follow my reasoning as I am confident in my working but I am just as convinced I am missing something fundamental.

Thanks for your help so far

Mike

ps. the graph x axis is plotted as a function of pi, i.e. 1 is actually pi, 0.5 is pi/2