What exactly is a constant angle?

Question

I previously asked a question about what a non constant angle is but it was closed due to lack of clarity and hence, I'm posting a new question. The notation $x^c$ will be used in this question to represent $x$ radians.

So, a proof in my Mathematics textbook is about proving that radian is a constant angle and it has been proved by expressing $1^c$ in terms of two constants, i.e. $1^c = \dfrac{\text {a straight angle}}{\pi}$.

As I understood it at first, a constant angle measurement is an angle measurement that does not depend on the length of it's initial or terminal sides but I realized that this is wrong as if some angle measurement depends on the lengths of initial or terminal sides, then it cannot be qualified as an angle measurement because an angle measurement is only supposed to measure the degree of rotation from the initial to terminal side which does not depend on the length of the sides.

User Somos stated that In many physical applications, an angle varies in time, and hence is not a constant angle. Think of a rotating propeller, for example. in my previous question. I didn't quite get what he meant. The angle might change with respect to time, for example, the angle b/n the arms of a compass may be $30^o$ at a point in time and $45^o$ at another, but that doesn't mean that under certain circumstances, the value of an angle of $30^o$ would change, right?
Did I misunderstand something in this?

Now, the two questions I have are :

What was the meaning of Somos' statement?

When we say that an angle measurement is constant, with respect to what factor are we declaring it constant?

Thanks!

Answer 1

I don't know what mathematics textbook you are using, but the angle equation is essentially relating two angles by giving their ratio which is $\,\pi\,$ which is also a well-known real constant. Of course, the two angles are also well-known constant angles themselves. In this context, the reference to the adjective constant seems not helpful to me. It does not add any useful information to the reader, and can be a potential cause of confusion.

In the context of physics, there are many physical constants such as the speed of light. These constants come with units of measure such as meters per second. Originally they were measured but more recently some of them are defined to have exact values. Refer to the Wikipedia article Metric system for details. One exceptional case is that of measure of angles.

The Wikipedia article Degree states

It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.[4] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.

So the measure of an angle is regarded as the ratio of two quantities with the same units although a selected angle can be regarded as a unit angle. This is the case for degrees and radians which have exact values.

Answer 2

A constant angle is a measure of rotation between two straight lines. It is independent of all measures of connecting or surrounding sides used to compute the angle.

The following angle definitions are free of measure of sides:

$$ \theta = L/R $$

$$ \cos A = \dfrac{b^2+c^2-a^2}{2 b c}$$

One full rotation of a segment of a line around an end as center is defined $2 \pi$, half rotation is $\pi$ and quarter rotation is $\pi/2$ etc.

In Euclidean geometry the angle is constant when seen with respect to a reference direction.Thus

An angle defines qualitative invariance of direction and quantitative magnitude of deviation from a fixed reference direction.

A straight line has a fixed direction with no curvature. When curvature changes in a curve, the angle to the reference direction changes from point to point.

In Physics:

During refraction when we say ratio of sines of angles of incidence and refraction is constant it is implied that the direction of each ray does not change.

In a rainbow optics.. if the angle between incident straight line and exiting refracted ray at a raindrop is $47^{\circ}$ then that angle is a physical constant angle associated with phenomena of total internal reflection and refraction... is a constant and can be called a property of a rainbow.