5 ms·
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
by rudiger 16y ago
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.
- amalcon 16y agoHow 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.
- aplusbi 16y agoIt'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.
- amalcon 16y agoThat'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.
- aplusbi 16y agoIt'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.
- amalcon 16y ago1 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.
- aplusbi 16y agoRadians 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).
- fooandbarify 16y agoDanger being, of course, that sin(1/4) != 1. (sin(0.25) = 0.247403959)
- amalcon 16y agoIf your sin function is calibrated to take radians, then yes. If it's calibrated to take cycles, sin(1/4) = 1.
- fooandbarify 16y agoAhh yes, of course. My apologies!
- sophacles 16y agoYou are exhibiting what I call "expert idiocy". 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".
- dunstad 16y ago>>>"...understanding that sin represents the y value of a point on a unit circle..." Wow, I'm enrolled in a college trig class right now and your comment just made me realize that. Thanks for the heightened understanding :)
- swolchok 16y agoFor 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. Wikipedia's picture is frightening (http://en.wikipedia.org/wiki/Unit_circle http://en.wikipedia.org/wiki/Unit_circle), but perhaps http://www.themathpage.com/atrig/unit-circle.htm http://www.themathpage.com/atrig/unit-circle.htm or http://www.snow.edu/jonathanb/Courses/Math1060/unit_circ_trig.html http://www.snow.edu/jonathanb/Courses/Math1060/unit_circ_tri... will help.
- aplusbi 16y agoWhat 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.