Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.
But for a while I have wanted to add just this one thought to the conversation and now is as good a time as any.
Some people advocate tau for pedagogical reasons, claiming it’s better for teaching radian measure of angles and also trigonometry. Meh. A good dose of cutting slices of circles is really a much better innovation than rewriting the books to use a different constant. Plus I have this one important point to make:
The values of the trig functions are calculated based on how far away the angle is from the nearest multiple of pi.
Look: Say I want to calculate the value of sin(3π/4). I draw the angle 3π/4 as a length around the unit circle and I remember that sin(3pi/4) is the y-coordinate of the point I have come to. Then I notice that this y-coordinate is the vertical side of a little triangle drawn down to the x-axis, and that therefore sin(3π/4) is the same as sin(π/4).
Did you see what happened there? sin(3π/4) was calculated by seeing how far 3π/4 is from π and using that distance instead! This is exactly how you do actually do it! The values of sin(t) and cos(t) are always calculated based on how far t is from the nearest whole multiple of π! You want cos(10π/3)? Well the nearest whole multiple of π is 3π, and we’re π/3 beyond that into the third quadrant, so cos(10π/3) = -cos(π/3).
I shudder thinking how much harder this would be if we were using tau and had to find out how far our angle was from a whole multiple of a fraction of τ! My university students have enough trouble with fractions as it is!
So there you go: I’ll keep using pi because you calculate the trig functions based on how far the angles are from the nearest whole multiple of pi. That’s all I have to say on the matter for today.