Trigonometry just the conversion factor of coordinate types?

[onethatyawns]

I think trig is assumed to be based upon triangles. This lumps trig next to squares, trapezoids, pentagons, hexagons, etc. Sure, triangles can be used to describe trig functions, but I think they do a disservice to your intuition. It's similar to Riemann sums versus integrals. True, integrals can be defined as tiny little rectangles, but I think focusing on this fact misses the bigger picture.

Trigonometry is merely the relation between angles and distances. Angles and distances are the two most primal types of coordinates/measurements. Some specific styles are more popular than others, such as Cartesian and polar of course, but all of the varieties rely upon only two things: angle and distance. Trigonometry is the study of the relationship between angle and distance.

Excuse my fluffy piece. I will shut up if people here don't like talking about the "why" of certain concepts. However, I think these things are important if one is to develop the highest level of intuition, and I think only the highest level of intuition is capable of breaking barriers between the status quo and the next discovery.

 

[axmls]

An integral is a Riemann sum--the limit of a Riemann sum (that is, if you're talking about Riemann integrals). And trigonometry is absolutely based off of triangles (trigonometry -- triangles). Note that you can use triangles to create squares, trapezoids, pentagons, hexagons, etc.

It turns out, triangles are quite useful objects. take any two points in a plane, and you can draw a line between them. In a Cartesian plane, you can break that line up into an x-component and a y-component--a right triangle. I don't think they're bad at all to intuition.

Of course, maybe I missed the point of what you're trying to say here.

 

[onethatyawns]

An integral is a Riemann sum--the limit of a Riemann sum (that is, if you're talking about Riemann integrals). And trigonometry is absolutely based off of triangles (trigonometry -- triangles). Note that you can use triangles to create squares, trapezoids, pentagons, hexagons, etc.

It turns out, triangles are quite useful objects. take any two points in a plane, and you can draw a line between them. In a Cartesian plane, you can break that line up into an x-component and a y-component--a right triangle. I don't think they're bad at all to intuition.

Of course, maybe I missed the point of what you're trying to say here.

An integral is the sum of points in a region. Using rectangles to describe how these are added is a bit simplistic.
Trig functions convert angles to distances and vice versa. Yes, a triangle is the simplest 2D shape which contains an angle and a distance (using Cartesian coordinates at least; I would argue circles are simpler than triangles). I think using triangles to be the primary explanation of trig is a bit simplistic.

When I say "simplistic", I actually mean less simplified. It is "simplistic" to me because it is a layperson's or simple person's explanation, and it shrouds the deeper, simpler meaning in secrecy.

I'm not trying to call anybody names.

 

[axmls]

By definition, an integral is the limit of a Riemann sum--i.e. just the infinite sum of a bunch of rectangles. That's really the gist of it when you get down to it. The Riemann integral is defined in terms of supremums and infimums of lower and upper sums (and the sums are just a width times a height).

A trig function doesn't turn an angle into a distance. Rather, it turns an angle into a ratio. This ratio is between the lengths of the sides of a triangle (the unit circle is a special case where the hypotenuse is $1$, so in this case the trig functions do give a distance, if you look at it that way).

 

[Mark44]

I think trig is assumed to be based upon triangles.

Well, of course. The word "trigonometry" means "the measure of trigons (or triangles).

This lumps trig next to squares, trapezoids, pentagons, hexagons, etc. Sure, triangles can be used to describe trig functions, but I think they do a disservice to your intuition. It's similar to Riemann sums versus integrals. True, integrals can be defined as tiny little rectangles, but I think focusing on this fact misses the bigger picture.

Trigonometry is merely the relation between angles and distances. Angles and distances are the two most primal types of coordinates/measurements. Some specific styles are more popular than others, such as Cartesian and polar of course, but all of the varieties rely upon only two things: angle and distance. Trigonometry is the study of the relationship between angle and distance.

Excuse my fluffy piece. I will shut up if people here don't like talking about the "why" of certain concepts. However, I think these things are important if one is to develop the highest level of intuition, and I think only the highest level of intuition is capable of breaking barriers between the status quo and the next discovery.

 

[Mark44]

An integral is the sum of points in a region.

No it isn't. It makes no sense to find the "sum of points." There are many interpretations of a definite integral, of which one of the simpler interpretations is the area between two curves and above some interval on the horizontal axis. The definite integral can be approximated by the sum of the areas of thin rectangles, with better approximations using more rectangles. We don't add up a bunch of points that have zero area.

Using rectangles to describe how these are added is a bit simplistic.
Trig functions convert angles to distances and vice versa.

More accurately, trig functions map angles to ratios. But how does this relate to integrals?

Yes, a triangle is the simplest 2D shape which contains an angle and a distance (using Cartesian coordinates at least; I would argue circles are simpler than triangles). I think using triangles to be the primary explanation of trig is a bit simplistic.

As we can determine from the name, trigonometry started off as the study of triangles, and triangles are at the heart of it. Once you venture past right triangle trig, it is possible to say things about angles that could not possibly be in a triangle.

When I say "simplistic", I actually mean less simplified.

Actually, "simplistic" means "too simple" in the sense of not considering all possibilities or parts.

It is "simplistic" to me because it is a layperson's or simple person's explanation, and it shrouds the deeper, simpler meaning in secrecy.

Perhaps you are really thinking of right triangle trig.