Question

\[arctan(x)=x-x^3/3+x^5/5-x^7/7+....\] Substituting x=1 into the equation, the right side yields 0.25 pi.
Where in the derivation of the Taylor expansion was the assumption that we are working in radians?

Answer 1

the assumption is that we are working in unitless quantities, no matter that they are radians
if we use a non-dimensionless quantity we get ugly constants outside due to the chain rule

Answer 2

argh, I am having a tough time making a good demonstration. The basic point is that calculus is done in pure numbers, and the fact that radians are unitless makes them good to deal with.
I will try to make some way of writing it to show how the constants pop up...

Answer 3

Are degrees not also unitless?

Answer 4

no, they are not

Answer 5

An explanation of why that is so would explain the original question well enough for me.

Answer 6

\[y^{\circ}=\frac \pi{180^{\circ}}x\]right? that is the conversion we would need to use everywhere if we wanted to switch from degrees to radians.

Answer 7

but what is a radian?

Answer 8

right, and what is the definition of pi ?

Answer 9

Circumference/radius and a few constants, thus dimensionless.

Answer 10

As length/length

Answer 11

Exactly. Also, if we use pi the number intrinsically has the representation of half a circle in some sense.

Answer 12

Given that a degree-rad conversion factor exists

Answer 13

well that would be true, though I'm not sure it was the aim of my point

Answer 14

yes, that is why I wrote the formula above without the word "radian" written in

Answer 15

360 is a purely arbitrary number, like seconds in a minute, or centimeters in a meter, 2*pi is a gemetrical relationship like tan, cos, etc

Answer 16

geometrical*

Answer 17

Whenever we introduce some arbitrary size of a thing ("I shall call this distance here one meter"), we need to use a unit. If we use a geometrical relationship, this is not an arbitrary relation; it is determined by mathematics. It has no need of units.

Answer 18

But why was it decided that x would be measured in the ratio of the diameter to circumference, and not radius to circumference, for example?

Answer 19

similarly with a circle, saying "I shall call once around the circle 360 degrees" is arbitrary, saying "2*pi relates the circumference (which represents a whole revolution, whatever you want to call it) with it's circumference is an inevitable mathematical consequence".

your question is a good one, doing such a thing would not change the formulas though

Answer 20

If we had defined Tau as the relation of a circumeference to it's radius, all formulas like\[e^{ix}=\cos x+i\sin x\]still hold just the same, but\[e^{i\tau}=\cos\tau+i\sin\tau=1\]

Answer 21

Inserting 1 into arctan(1) gives (45 degrees/ 0.25 pi radians or whatever). It's just curious that the angle system that has a conversion factor of 1 with the answer given is the radian.

Answer 22

I suppose that is the essence of the question.

Answer 23

here's one that may help from yahoo answers:
Look at how it is shown that the derivative of sinx is cosx. You will find that the limit of sin(h)/h as h goes to 0 is needed. For radian measure, this limit is 1. For degrees, this limit is pi/180. If you look closer, the derivation uses the fact that the area of a circular sector is (1/2)r^2 (theta), where theta is the radian measure of the angle. If degrees are used, the formula is (pi/360) r^2 (theta). If you look at how the area formula is derived, you should be able to see why radians are used.

Answer 24

So it is in the differentiation of tan that radians were implicit. Excellent.

Answer 25

Yeah I told you before, but couldn't show it. There are other reasons too that I tried to convey above.

Answer 26

I am not really articulating the answer well I feel :(

Answer 27

I got what you were trying to say earlier, but the appeal to fundamentality in itself seemed a little trivial.

Answer 28

I know you already got to an answer, but maybe this will help.
The definition of radian is the lenght of the arc when it has radius=1.

The tangent is the lenght of the projection of the point as shown in the figure when the circle has radius=1.
When talking about circles with radius 1, we can then say that those projections have unit of lenght, and therefore the function tangent must take the lenght of the arc in the lenght of the projection. If we use degrees, the units of the coordinate axis need to change so that the the projection still transforms lenght in lenght, but we don't do this when working with tangents, therefore, the inverse of the tangent mus take that lenght in another lenght and since we work with a certain unit when using tangents, the unit of the angle must be the same.
Maybe this to visualize.