Omg Sin And Cosine Have Been Eliminated

[MfA]

Well, there's no real need to even calculate final numbers. They could leave everything as radicals and trig functions and leave it at that.

It's sloppy work if all the trig functions cancel out. Having say an entire line filled with a long convoluted formula as an answer, when the real answer is the square root of X betrays a lack of understanding of the math IMO. It won't evaluate as correctly as the simpler answer either, more ways for numerical errors to sneak in.

Let the students use whatever tools for the job they need to solve the problem. Ultimately, it's about solving problems, not how good one is at arithmetic.

Depends if you are trying to teach them engineering or math. Existing symbolic solvers are tools too, knowing how to present them with a formulation of a problem they can solve won't necessarily help you understand how they actually solve it.

 

[Fred]

There are many ways to define sin and cosine, ti sorta depends on which branch of mathematics your proffessor is a specialist in (they all claim theirs is the most fundamental)

You can define them in terms of their expansions around some neighborhood (a taylor series for instance) or you can look at their algebraic relations. You can also somewhat more abstractly run them backwards as fundamental representation elements of the special orthogonal groups etc etc. Or you can look at them as solutions of some differential or integral equation.

Whatever the axioms their existance and uniqueness is not in doubt, as such this bizarre media frenzy is just another in the long line of head scratchers.

 

[KimB]

There are many ways to define sin and cosine, ti sorta depends on which branch of mathematics your proffessor is a specialist in (they all claim theirs is the most fundamental)

You can define them in terms of their expansions around some neighborhood (a taylor series for instance) or you can look at their algebraic relations. You can also somewhat more abstractly run them backwards as fundamental representation elements of the special orthogonal groups etc etc. Or you can look at them as solutions of some differential or integral equation.

Often I've wondered how anything can be a more fundamental definition of a function than where it originally came from historically. Consider two things:
1. Defining a sine via a series: this requires that one define a particular series. But where did this series come from? Well, it came about by discovering what series fit the sine function, so this definition is just circular.
2. Defining sine as a solution to a differential equation: this would at first sound a bit better, but sin(x) typically appears as a solution to a second-order differential equation along with cos(x), and in this situation one need not use sin/cos at all in specifying a complete solution to the differential equation. And thus this situation arises again: sin/cos arise as the solutions of differential equations because we recognize them as such, not because they are the most natural solution to select.

So, I'd claim that the definitions of sin/cos as anything but the ratio of the sides of a triangle is a circular argument. But the fact that they are used for other things really shows that they cannot be done away with through recasting variables as in rational trigonometry.

 

[MfA]

And thus this situation arises again: sin/cos arise as the solutions of differential equations because we recognize them as such, not because they are the most natural solution to select.

Hmm? If some math genious figured out calculus without ever thinking about trigonometry it wouldn't change a thing about the solutions ... he would name them differently, but they'd look the same on a graph.

As for their definition in geometry, it is far from a natural one. That the relationship between angles defined to be linear along a circle and the sides of a right triangle could be used to solve problems with arbitrary triangles was a pretty big leap ... and without that leap such a definition of angles and their relation with right triangles is pretty arbitrary and irrelevant. Harmonic oscillations turn up everywhere, it's occurence in trigonometry is as haphazard as anywhere else.

Hell, in the end we actually need them for calculus ... whereas you only need a couple of algebraic relations between trigonometric functions to solve trigonometric problems (you don't actually need to have a calculable definition). Kinda funny that you don't need the actual trigonometric functions for trigonometry.

 

[DemoCoder]

It's sloppy work if all the trig functions cancel out. Having say an entire line filled with a long convoluted formula as an answer, when the real answer is the square root of X betrays a lack of understanding of the math IMO.

No one's saying that they don't have to simplify formulas, only that plugging in numerical constants and coming up with a single number at the end is not the point of the exercise. Students need to be taught to think abstractly, not to waste 50% of their time in exams doing addition, multiplication, and division on paper.

When I took trigonometry, it was co-taught the same year with calculus. I see no need to *dumb down* curricula for the sake of teaching trig sans trig functions and square root so students don't need calculators or trig tables when in the real world, they are going to encounter trig functions all over the place.

Depends if you are trying to teach them engineering or math. Existing symbolic solvers are tools too, knowing how to present them with a formulation of a problem they can solve won't necessarily help you understand how they actually solve it.

You can understand how they solve it perfectly without plugging in numbers and doing arithmetic. And if you are teaching an engineering course like statics or dynamics, having them plug in and do arithmetic is even worse. I don't know how they teach Newtonian dynamics in other countries, but when I took it, we weren't dealing with single digits, but numbers with 3 or more significant digits, where doing the arithmetic by hand would take up so much time during an exam you'd be left with little else.

 

[KimB]

Hmm? If some math genious figured out calculus without ever thinking about trigonometry it wouldn't change a thing about the solutions ... he would name them differently, but they'd look the same on a graph.

Sure it could change the solutions. The differential equation that sin/cos functions are often quoted as the solutions of is the equation:
(d/dx)^2 f(x) + C * f(x) = 0
...and there are many ways to formulate the result, since any linear combination of solutions will be a solution. We often use sin/cos functions as solutions because of the history of sin/cos functions. There isn't any other reason than historical that we use them as solutions.

A more obvious solution to the above differential equation is actually:
f(x) = A*e^(ikx) + B*e^(-ikx)
...where k = 1/sqrt(C) and A and B are any two constants.

As for their definition in geometry, it is far from a natural one. That the relationship between angles defined to be linear along a circle and the sides of a right triangle could be used to solve problems with arbitrary triangles was a pretty big leap ... and without that leap such a definition of angles and their relation with right triangles is pretty arbitrary and irrelevant. Harmonic oscillations turn up everywhere, it's occurence in trigonometry is as haphazard as anywhere else.

My point was that the definition in geometry is directly tied to its history. So I think it makes the most sense as a way to define the sin/cos functions, most particularly when introducing the concept to new students.

Hell, in the end we actually need them for calculus ... whereas you only need a couple of algebraic relations between trigonometric functions to solve trigonometric problems (you don't actually need to have a calculable definition). Kinda funny that you don't need the actual trigonometric functions for trigonometry.

Well, actually, if you decide to take the leap to complex numbers there is no need for sin/cos functions in calculus at all, most particularly since:
sin(x) = (e^ix - e^-ix)/2i
cos(x) = (e^ix + e^-ix)/2

Edit: Or, more compactly:
sin(x) = Im(e^ix)
cos(x) = Re(e^ix)

 

[Fred]

Historical context need not be 'general' context. For instance, who made euclidean geometry special all of a sudden? Its anthropogenic by nature (and in fact physically not even applicable since 1905).

For instance the usual polar variables r and theta can just as well be replaced by hyperbolic coordinates u and v, and in fact the latter are more general and 'natural' in certain branches of mathematics.

Moreover you claim power series expansions are less relevant to define sine. I say in a sense they are more fundamental and 'general' a notion, even the very particular one that corresponds to sine.

There exists topologies with very limited notions of what 'length' means, but which do admit local series expansions. Indeed the only notion of what an angle is, corresponds exactly to that sine expansion.

The point is, the definition depends entirely on what you are working on, there is no prefered axiomitization in mathematics, only what is most interesting.

 

[DemoCoder]

More most useful.

I can solve alot of problems in fields that are isomorphic to the integers or reals, but some constructs make some problems easily to state or express than others.

But I think we are missing the point of this thread: The idea that the preferred constructs are those where you don't need calculators. Arithmetic is a commodity, a strictly mechanical skill, easily mastered by idiot savants, easily automated, and thus not something we need be overly concerned with. Students, after learning the basics of arithmetic, should be allowed to "outsource" these computations to devices, as surely as we allow them to use a protractors, an abacus, or even pen and paper.

Now if the argument is, pick the construct that allows simpler algebraic manipulation, fine. But I find the calculator argument bogus.

 

[KimB]

Well, one could make a similar argument about algebraic manipulation that you just made about arithmetic, actually. The more challenging connection, I think, typically comes when you attempt to link mathematical equations to the real world. That, at least, is the problem that manifests itself the most in lower-division physics classes.

 

[DemoCoder]

The only difference is, it is impossible to mechanize all algebra (e.g. proofs) in a consistent way. Mathematica, Maple, et al are powerful tools, but they have severe limitations which can not be overcome even in theory.

So sure, one could use a tool like Mathematica to code up what-if searches for problems, to graph stuff, and to explore structures. And in this regard, Mathematica is a powerful teaching tool. I would have no problem with a student using Mathematica to simplify an expression.

But Mathematica won't solve your problems for you. So you still need to know algebra, at the very minimum, to even cast your problems in Mathematica format, and if you need to prove something, your SOL in most cases. Automated theorem provers are hard as hell to use and require almost as much work to setup as solving the problem manually, and they often produce bad results without step by step "help" from the human being.

The fact is, arithmetic is mechanistic. Symbol manipulation however, by term rewriting, depending on your language domain, is not decidable mechanistically in general. Some of it can be automated, but you cannot escape the fact that the automated systems will fail.

So in this regard, arithmetic and algebra are not equivalent.

 

[Simon F]

And thus this situation arises again: sin/cos arise as the solutions of differential equations because we recognize them as such, not because they are the most natural solution to select..

They are (arguably) the most natural ones to select.

There are a family of solutions to the problem, e'(x)=f(x), f'(x)=e(x), but the logical (non-trivial) choice is the solution that has e(0)=0 and e'(0)=1.

 

[KimB]

Er, I'm sorry, I was considering a different differential equation:
d^2 f(x)/dx^2 + C*f(x) = 0

...which is a typical wave equation where e^ix and e^-ix are more natural solutions than sin/cos.

But still, the differential equations you're describing typically arise from situations of rotational motion, and as such both the differential equation and the most natural solution may look entirely different if a different coordinate system is chosen (as is frequently done for planetary orbits, for example).

 

[MfA]

Non damped mass spring systems aren't exactly uncommon either.

But I think we are missing the point of this thread: The idea that the preferred constructs are those where you don't need calculators.

Part of it, another part is that the preferred constructs are those which easily provide an exact answer. The square root of 2 is a better answer than 1.414 IMO.

Now if the argument is, pick the construct that allows simpler algebraic manipulation, fine.

Yes, well ... what of that arguement then?

 

[KimB]

Non damped mass spring systems aren't exactly uncommon either.

Right, but the differential equation that one obtains in those systems is the aforementioned second order differential equation, where I claim that the complex exponential solution may well be the more obvious (and easier to work with), to somebody that is familiar with complex numbers and exponentials.