Perhaps you're right about people doing "actual" complicated maths simply not caring about a multiplicative constant. However, a lot of people do most or all of the complicated maths they're ever going to do in their lives when they're in university. During this period, the correct symbol makes every formula and equation simpler and easier to learn. Crucially, it's also during this period that people's understanding of complicated maths is most important, as they are judged by letter grades upon which many of life's opportunities depend.
How often do people actually make this error? I make all sorts of errors of that class when doing math by hand, but I've never once been off by a factor of two because the formula calls for 2pi.
It's not about errors but understanding. A lot of math involving trig is abstract enough to be confusing to most people. Tau makes is [slightly] less so. For example, understanding that sin represents the y value of a point on a unit circle is easier when 1 tau is a full circle rather than 2 pi.
For the sake of completeness, cos is the x value, because of the identity sin(x)^2 + cos(x)^2 = 1. You should find a good unit circle trigonometry picture if your class isn't giving it to you.
What I find especially ironic is that while I completely understood that I could get the x and y values using rcos(t) and rsin(t) and used it a lot when I worked for a video game company, it's only more recently that the idea "clicked". I always thought about sin and cos in relation to triangles and not circles.
That's only a problem when you need to use radians, which is only necessary for calculus. You can do all the explanations with "cycles" (1 cycle = 2*pi radians), and it's actually simpler.
It's not simpler, it's identical. 1 "cycle" = 1 tau = 2 pi.
And in any case, I think using degrees to teach trig is a terrible idea and only causes more confusion later. sin90 = 1 makes no sense, and the fact that that's my first thought when thinking about sin only causes me problems. If I had instead been taught radians first, a lot of stuff would be significantly easier.
1 cycle = 1 "tau" radians. It's important to include units. I don't see how keeping irrational "magic numbers" out of the equation could make it anything but simpler. As an added benefit, it would serve as a transition between degrees and radians, by introducing the idea that there are multiple measures for an angle without simultaneously introducing a unit that has irrational values in every useful situation.
I agree that degrees are terrible, but sin(1/4)=1 makes a lot of sense. Probably even more than sin(1/4 tau)=1. The only reason to use radians instead of cycles is that changing the units breaks the wonderful trig derivative symmetry.
Radians are pure numbers (which is why I left them out). That said, I see what you mean now and I agree. sin(1/4 of a circle) makes more sense than sin(1/4 tau) which makes more sense than sin(1/4 360).
Basic teaching theory states that experts don't think like novices, and further, they are likely to not remember the misunderstandings and difficulties they encountered as novices, because the novice problems are what they now consider simple. This is a being studied a lot in the world of education.
I call it expert idiocy when the expert refuses to accept that his understanding is actually a pretty advanced state of thinking, and not immediately obvious to the beginner. It is this form of idiocy that leads to people feeling that "only freaks can get math" or "I'm not smart enough for physics" or "computer geniuses can only do basic tasks".
Perhaps you are correct. I don't remember the order in which I learned various concepts but I probably learned about degrees before pi and certainly before trig.
Since radians involve irrational numbers I can understand it being more difficult to learn than degrees. However degrees are a completely arbitrary unit that are used for historical reasons.
Getting stuck thinking in degrees hindered my ability to understand trigonometry, and I don't think I'm alone. It is my belief that teaching trig using radians would be less confusing than teaching trig using degrees first, then radians (as was my experience). With degrees there is no direct correlation between 90 degrees and the values of the trig functions which leads to people simply memorizing values. The same is true (to a lesser extent) when using 2 pi.
True, but you are now setting up a false dilemma. The point is not that arbitrary units are hard, nor that any 2 are relatively harder. It is that a consistent unit that is not arbitrary is easier than the arbitrary ones, and usually involves quite a few useful "symetries" (by which I don't mean real symetries, but conceptual linkages).
As for why people think degrees is easier: it is simply because common usage of degrees makes the concept familiar to learners. I totally agree that teaching trig in terms of radians first would be much better.
