An AoPS Problem for Half Tau Day 2025
Happy Half Tau Day, everyone! Today on 3/14 we celebrate the halfway point ’round the unit circle, τ/2 = 3.14159…
Tau, as you’ll likely recall, is the circle constant defined by the equation τ = C/r, a notation first publicly proposed by yours truly in 2010 via the now-infamous math essay The Tau Manifesto. Since radian angle measure is defined by θ = s/r, setting the arc length s equal to the full circumference C shows that θ = C/r = τ is the angle of a full turn of a circle. Setting r = 1, we see that, equivalently, we can think of τ as the circumference of a unit circle. It’s all so easy and natural with τ!
In honor of this semi-auspicious day, I thought I’d share with you an exercise (and a preliminary example) from a two-volume set of books I recently finished studying: The Art of Problem Solving, Volumes 1 & 2. Though nominally aimed at high-school students, AoPS (as it’s abbreviatedly known) is full of material sufficient to challenge even quite experienced students of mathematics.
The full post is fairly involved and involves a lot of mathematical typesetting not well-supported by email clients, so as usual I’ve published the full version on my website at michaelhartl.com:
The short of it is that we can use τ and a little elementary number theory to find a compact and elegant formula for the period of the sum of a sine and a cosine. It is exactly the kind of problem I wanted to know how to do when I started studying The Art of Problem Solving, and which (despite having a Ph.D. in physics) I couldn’t do even a few months before encountering the problem in 2024. This was due not to any deficiency in my understanding of periods of sinusoids, but rather to a weakness in my understanding of elementary number theory.
I hope you enjoy it!
Cheers,
Michael