Quick random questions to help me clear stuff up

[lamball1]

Quick random questions to help me clear stuff up :)

Hey
I'm studying for my final exams and oh my have I forgotten a lot of things.
so I have a few questions, answer them If you wish and/or can:

If f $\propto$ a and a $\propto$ 1/m
why can we say that a $\propto$ f/m ??

secondly, I have never understood why only radians can be used in calculus...

thirdly, I'd like to understand why the chain rule works a little better...

thx

 

[Mute]

Hey
I'm studying for my final exams and oh my have I forgotten a lot of things.
so I have a few questions, answer them If you wish and/or can:

If f $\propto$ a and a $\propto$ 1/m
why can we say that a $\propto$ f/m ??

See this post.

secondly, I have never understood why only radians can be used in calculus...

You can use degrees, but $\sin x$ and $\cos x$ and the other trigonometric functions are defined using radians as their arguments, so converting radians to degrees introduces a proportionality factor that you have to remember to account for when you take derivatives or do integrals, etc. Radians are chosen as the standard measure of an angle because they are a dimensionless measure of an angle defined in terms of the properties of a circle; namely, the angle in radians subtended by an arc of length $s$ on a circle of radius $r$ is just $\theta = s/r$.

 

[HallsofIvy]

If x were in degrees then the derivative of sin(x) would be $(180/\pi) cos(x)$ while if it is radians the derivative is just cos(x).

But, in fact, I would say that the x in sin(x) or cos(x) does not have units of either radians or degrees- it is not an angle and does not have any units at all just like in $x^2$ or other functions. There are a variety of ways to define the sine and cosine functions so that the argument does not have any units at all. One is to define it on the unit circle: Starting from the point (1, 0), on the unit circle, measure distance t around the circumference of the circle (counterclockwise for positive t, clockwise for negative t). "Cos(t)" is defined as the x-coordinate of the final point, "sin(t)" as the y-coordinate.

It is also possible to define "y= cos(x)" as the function satisfying the 'initial value problem' y''= -y, y(0)= 1, y'(0)= 0 and define y= sin(x) as the function satisfying y''= -y, y(0)= 0, y'(0)= 1. Or define cos(x) as $\sum_{n=0}^\infty (-1)^nx^{2n}/(2n)!$ and sin(x) as $\sum_{n=0}^\infty (-1)^n x^{2n+1}/(2n+1)!$

 

[lamball1]

Some of it is waay over my head but I got the proportionality though. thx.

"Radians are chosen as the standard measure of an angle because they are a dimensionless measure of an angle defined in terms of the properties of a circle"

why aren't normal degrees dimensionless considering that they're also based on property of a circle (angle for circumference/360)??

 

[arildno]

Some of it is waay over my head but I got the proportionality though. thx.

"Radians are chosen as the standard measure of an angle because they are a dimensionless measure of an angle defined in terms of the properties of a circle"

why aren't normal degrees dimensionless considering that they're also based on property of a circle (angle for circumference/360)??

No one denies that normal degrees aren't dimensionless.

Normal degrees, however, and every other angular measurement system other than radians are arguably more COMPLEX than radians, in that radians are strictly equal to the ratio s/r, where "s" is arc length", "r" "radius.
The other angular measurements boils down to different systems of k*s/r, where k is not equal to 1.

Radians are conceptually simplest.

 

[lamball1]

thx, I get it