This is a contribution question I'm making in hopes that others may benefit. I will provide my answer underneath. Initially I wanted to ask this question, but I solved it myself and I'd like to give back for the question I asked earlier. I will wait a day before selecting an answer in case anyone else wants a crack at it.
I asked a question earlier here about why I was getting a different answer when calculating the tangent slope of a line by using the secant method compared to using the Calculus method. I found out that the Calculus method always produces an answer relative to radian measure.
So the two answers I had were actually the same, except one was slope per radians and one was slope per degrees.
Why is it that when we convert radians to degrees we multiply radians $\ *\frac{180}{\pi}\ $, but when we convert slope per radians to slope per degrees we have to multiply the inverse conversion formula slope per radians$\ *\frac{\pi}{180}\ $?
I should give an example to explain better.
Given r radians, we get degrees by $\ r * \frac{180}{\pi}\ $
Given d degrees, we get radians by $\ d * \frac{\pi}{180}\ $
However, the opposite is true when converting slope per radians and slope per degrees:
Given s slope per radians, we get slope per degrees by $\ s * \frac{\pi}{180}\ $
Given t slope per degrees, we get slope per radians by $\ t * \frac{180}{\pi}\ $
It seems illogical that we convert slope per radians to slope per degrees by using the formula which converts degrees to radians. Since it's already in radian form?
Best Answer
If we want to know an hour in terms of minutes, we multiply 1 hour $\times \frac{60}{1}$, given the result in minutes. If we want to know how convert 180 minutes, we divide $180$ minutes by $60$, i.e., multiply $180 \times \frac 1{60}$.
You'll find this phenomenon in any conversion: To convert temperature in degrees Celsius to temp in Fahrenheit, we have $F = \frac 95 C + 32$. To convert to from F to C, we need to invert this: $C = \frac 59(F-32)$
$R \text{ radians}\;\times \dfrac{180^\circ}{\pi \;\text{radians}} = \dfrac{R\times 180^\circ}{\pi}$.
"Radians" cancels as the unit, leaving a numeric value expressed in degrees.
$D \text{ degrees}\; \times \dfrac{\pi \;\text{radians}}{180\; \text{degrees}} = \dfrac{D\pi\;}{180}\;\text{ radians}$.
"Degrees" cancels as the unit, leaving the value expressed in radians.
Moved from comments:
Note that slope per radian is a ratio: $\;\dfrac{\text{slope}}{\text{radians}}.\;$ So to obtain "slope per degree", you need to have $π$ radians in the numerator, to cancel the unit "radians" from the denominator, and $180$ degrees in the denominator, to end with slope/degrees.
Slope itself is not a "unit" per se, meaning it isn't a degree, or radian, a mm, or foot. It is unit-free: even if we assign distance units (meters, say) to displacement: e.g. $Δ\,y\text{m}=y_2\,\text{m}−y_1\,\text{m}$ and $Δ\,x\,\text{m}=x_2\text{m}−x_1\,\text{m},$ then we have $$\text{slope}\,= \dfrac{Δy\,\text{ m}}{Δx\,\text{m}}=\frac{Δy}{Δx},$$ you see that the units "m: meter" attached to "change in y" and "change in x" cancel in the ratio defining slope, leaving us with a unit-free scalar which slope really is.