$.0030077$ radians is $\arctan(.0030077)$ degrees?
In this question Related Rates Hot Air Balloon, the op converts from radians to degrees, by taking the $\arctan(\text{radians})$. This is obviously a wrong method. In fact if your calculator is set to degrees then $\arctan$ will give you an answer totally off. But if your calculator is set to degrees it gives a good approximation for $x \approx 0^+$:
So if you just find the $\arctan$ of your radians you will get a close enough answer if your calculator is set to degrees.
I believe this has something to do with:
$$\lim_{x \to 0^+} \frac{\arctan (x)}{x}=1$$
For $x \approx 0^+$
$$\arctan (x) \approx x$$
Hence for $x$ near $0^+$:
$$\arctan (\frac{180}{\pi}x) \approx \frac{180}{\pi}x$$
In which case if your calculator assumes $x$ to be in degrees when calculating $\arctan (x)$ so it converts it to $\frac{180}{\pi}x$ then it it gives you back $\approx \frac{180}{\pi}x$ which is the expression for radians if $x$ is degrees.
Is this correct. I'm don't know much on this so I'm not trying to give any false information.
While angles can be measured in radians as well as in degrees the function $\arctan$ accepts pure real numbers (e.g., slopes of lines in the $(x,y)$-plane) as inputs and gives an angle, measured in radians as output. If $|x|\ll1$ then $\arctan x\approx x$. It follows that $$\alpha:=\arctan(0.0030077)\approx 0.0030077\ ,$$ whereby the value on the right hand side is the value of $\alpha$, expressed in radians. You can convert this angle into degrees as follows: $$\alpha={180\over\pi}\>0.0030077\ ^\circ=0.17233\>^\circ \ .$$