I had three reactions:
1) I was elated to finally understand sin and cosin.
2) I was thankful for him and also amazed at how simple it was in 5 minutes to explain something that seemed to have eluded me for 5 years of highschool. It made me wonder alot what other difficult subjects could be easily taught by just a good explanation.
3) I wondered why math class had failed so specifically to give me that knowledge, or at least test me to see if I understood it and if I didn't to ensure that I did understand it. The answer I think was math class just encouraged me to use a calculator, thus short circuiting any real understanding.
Anyways he went on to be a physics professor at Cambridge.
I always appreciated that moment.
I didn't understand what functions are/ how they work, my brain absolutely refused to understand, until i started programming. Then i got it in 5 minutes.
I don't know if it has to do with how my brain works. I wouldn't say i have too technical mind.
It's subtle, but I think it's a very valid point since I too never thought it in terms of ratio, merely in terms of here's the angle, this is what you need to find, you apply sin/cos/tan and get the answer.
It's possible back in the old days the math teacher did mention it, but we couldn't understand it at the time because we had little sense of the subject.
A common take on this phenomenon is discussed here occasionally:
http://habitatchronicles.com/2004/04/you-cant-tell-people-an...
For a demonstration, go back and watch a good movie that requires focus, say three times in a row. Every viewing should uncover details you didn't notice the first time, because we can't take them all in at once.
It's also why repetition is a good learning technique. Unfortunately few people have the patience to start over from the absolute beginning.
At your current age, you are more capable in some ways, less in others, and thus your learning capacity is different.
If you somehow managed to skip trigonometry in high school but still got into college calculus, then there are a few of other ways they might be defined.
One way is to put off dealing with trig functions until you get to differential equations. Then sin is the unique solution of y'' + y = 0, y(0) = 0, y'(0) = 1, and cos is the unique solution of y'' + y = 0, y(0) = 1, y'(0) = 0.
Another way is to define them axiomatically. These are the axioms for sin and cos:
1. They are both defined everywhere on the real line,
2. cos(0) = sin(pi/2) = 1, cos(pi) = -1,
3. cos(y-x) = cos(y) cos(x) + sin(y) sin(x)
4. for 0 < x < pi/2, 0 < cos(x) < sin(x)/x < 1/cos(x).
From that you can deduce all the usual trig identities and the limits you need to do calculus with them.
You still need to prove that functions satisfying those 4 axioms actually exist. One approach is to put that off until later when you've got more tools under you belt. For example once you do Taylor series, you can work out what they Taylor series would be for sin and cos if they exist. The functions defined by those Taylor series do exist, so then you just need to show that those functions satisfy the axioms.
But all of these require calculus and quite a lot of hand waving about real numbers and limits and infinite sums, especially in a high school context. And most of their applications (per se) are pretty advanced.
At the high school level it’s probably best to explain that angle measure per se is a very tricky concept (an angle measure is a type of logarithm). We can get a very long way without ever having any explicit "angle measure" concept. E.g. there are no angle measures anywhere in Euclid’s Elements. (Angle measures come from astronomy and navigation.)
We can look at the complex coordinates of 2D rotations (points on the complex unit circle) without ever trying to turn that into a single scalar-valued quantity. And should do that in a lot of circumstances where angle measures are currently used. In e.g. geometric simulations (computer graphics, robotics, computational geometry, ...) it’s best to avoid angle measures as much as possible, and stick to vector algebra.
An angle measure "is" a logarithm of a planar rotation with the orientation stripped away. That is, the structure of angle measures is isomorphic to these logarithms. (Planar rotations are most naturally represented by a quantity like a + Ib, where I is a unit bivector and a and b are scalars with a² + b² = 1.)
Logarithms need comparable amounts of technical machinery to define as arclengths, so there’s no difficulty advantage in considering an angle measure to be an arc length or the area of a sector instead of a kind of logarithm. Conceiving of a logarithm of a rotation as being the same quantity as the area of the associated circular sector can be a helpful mental picture though (and there is no need to even strip out the bivector’s orientation if you use an area).
You can calculate the length of an arc associated to a triangle by simple comparison to a known angle such as the quarter angle and its products, this will allow you to finitely compute any rational angle. This is the conceptual basis for using arclengths.
To formalize this, you can define trigonometric functions now, and you can use them to introduce polar coordinates, and by translation of the intersection to the origin, show that you can calculate the angle for all real numbers by construction of a right angle triangle and now determine the sine as the ratio of two real numbers, etc...
It is wholly unnecessary to introduce logarithms. Yes they are isomorphic. They are also unnecessary. So are bivectors. Angles are perfectly well defined without anything but basic algebra and some geometry. You really just need an extensible definition of the trigonometric functions to provide rotation and you've got yourself a perfectly sound definition.
And note that an "angle" and an "angle measure" are two separate concepts.
> finitely compute any rational angle
Sure, we can calculate specific logarithms without a formal definition as a function over a continuous domain, based on some stated rules for formal manipulation, taken as axioms. Not just rotations, but also scalar logarithms. For example, we can precisely calculate the logarithm base 2 for any arbitrary rational power of 2: log₂ 2^(p/q) = p/q.
A "rational angle" is nothing but a rational power of –1: Arg (-1)^(p/q) ≡ (p/q)π (mod 2π)
The rational angles are only used to introduce the trigonometric functions. From then on we can deal with any real angle using arcs of circles and trigonometric functions.
Therefore we don't have any issues. We can deal with angles perfectly well as long as we can know the circumference of a circle.
Whereas otherwise we would need either calculus or complex numbers, but we need geometry to make sense of them and we need angles to make sense of geometry without handwaving or making it needlessly difficult to learn.
In other words, an angle measure isn't a logarithm, a logarithm is equivalent to an angle measure when we accept constructs that are not necessary but are sufficient to angle measures.
This whole technology is built on a foundation of calculus (limits, infinite sums).
> we need angles to make sense of geometry
Euclid’s version of geometry involves no angle measures whatsoever. Vector methods (of the type which should be preferred for most geometrical modeling and computation) generally need no angle measures.
Angle measures get useful when dealing with uniform circular motion: astronomy and navigation, signal processing, etc., but are largely superfluous for geometry per se.
The responsibility for not knowing until that point is more likely to be on my shoulders than the lecturer's, however.
such a shame. sounds like a talented fellow who really could have made something of his life. maybe even a chair at Oxford :/
Why? How are they different?
Explain?
Overall what I've found is that older textbooks (edit: not just math) are more likely to (1) explain concepts in smaller steps; (2) not bury them in symbols and abstractions; (3) introduce and explain physical intuitions for subjects when they are relevant.
to learn anything, so you can teach it to somebody else (the REAL def of Learning I'd say), you have to Do The Exercises.
---- I recently went over one of my older books on Matrices and it was a curious mix of trying to be sort-of precise, with trying to bring the 'wonder' and 'magic' of the concepts to life early -- so as to maintain reader interest. Ultimately, it was the way of thinking (mathematically) that was mostly taught; the 'matrix intro' was just fodder to support the pedagogy!