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The reasonable ways to rigorously define sine and cosine functions (as functions) are: (1) (analytic continuations of) inverses of particular integrals (2) the

The reasonable ways to rigorously define sine and cosine functions (as functions) are: (1) (analytic continuations of) inverses of particular integrals (2) the sums of particular power series (3) solutions to a differential equation (4) as odd/even parts of the complex exponential, itself defined via (1), (2), or (3). Proving that these are all equivalent is a good exercise for a complex analysis student. 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.

sudosysgen 5y ago
You don't need logarithms for angle measures. Arcs of circles work pretty well too.
- jacobolus 5y ago
  Measuring arclength (in mathematics) requires calculus: that is, you need to take ∫|ds| = ∫√(dx² + dy²) or similar. It is inherently a tricky advanced concept which can’t be discussed in any precise way until after at least one calculus course. 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).
  - sudosysgen 5y ago
    I fear you may be overcomplicating the matter. If you want to measure an angle on the plane defined by two lines, simply form a circle of a radius one and take the ratio of the length of the arc at the angle to total circumference. 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.
    - jacobolus 5y ago
      You are implicitly dealing with a logarithm whether you say so or not. An angle measure is a logarithm. It’s a way of taking a naturally multiplicative concept (composition of rotations) and making a change of variables to convert multiplication to addition. 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)
      - sudosysgen 5y ago
        I'm dealing with something that is equivalent to the logarithm, sure. But since I'm not using anything beyond real numbers, this is infinitely simpler than bivectors or complex logarithms and I don't need calculus. 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.

sudosysgen 5y ago

You don't need logarithms for angle measures. Arcs of circles work pretty well too.

jacobolus 5y ago
Measuring arclength (in mathematics) requires calculus: that is, you need to take ∫|ds| = ∫√(dx² + dy²) or similar. It is inherently a tricky advanced concept which can’t be discussed in any precise way until after at least one calculus course. 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).
- sudosysgen 5y ago
  I fear you may be overcomplicating the matter. If you want to measure an angle on the plane defined by two lines, simply form a circle of a radius one and take the ratio of the length of the arc at the angle to total circumference. 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.
  - jacobolus 5y ago
    You are implicitly dealing with a logarithm whether you say so or not. An angle measure is a logarithm. It’s a way of taking a naturally multiplicative concept (composition of rotations) and making a change of variables to convert multiplication to addition. 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)
    - sudosysgen 5y ago
      I'm dealing with something that is equivalent to the logarithm, sure. But since I'm not using anything beyond real numbers, this is infinitely simpler than bivectors or complex logarithms and I don't need calculus. 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.