Because those sinusoidal waves are merely projections (in the real and complex plane) of the true elemental component of the FT, the complex exponential, which is really a helix: http://m.eet.com/media/1068017/lyons_pt2_3.gif

A while ago I made some drafts for a series of diagrams to help visualize this (http://imgur.com/vEcnVdn) but unfortunately never got around to finish it (https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustr...).

Anyway, for those who find it easier to think of sinusoidal curves, the animation in the Wikipedia article (https://commons.wikimedia.org/wiki/File:Fourier_transform_ti...) is a very good visualization (also, BetterExplained's rant on sines being explained as circles may resonate: http://betterexplained.com/articles/intuitive-understanding-...)

To be fair, the expression does contain (edit: ) Euler's Formula, which is basically drawing circles in the complex plane, so people are going to draw circles.

I started thinking about sinusoids this way after seeing stuff like this: https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAAC... — if you look down at the complex plane, you will see a circle.

Sine an cosine are functions which tells you what your coordinates are when you travel around a circle (of diameter 1) so anything containing those (here hidden in Euler's formula) will provoke thinking in terms of circles. I find it very useful btw. Once you realize sin(x) represents how high you are on a circle and cos(x) how far to the right (if you start walking on it from [0,1] counter clockwise) then a lot of stuff start making intuitive sense.