Why transform degrees into radians when computing linear approximation to find $\tan{44^\circ}$?

Question

I am asked to find the linear approximation of $\tan{44^\circ}$. Why should I transform degrees into radians to do that?

I understand that using degrees would give me a wrong solution (which would be $-1$ instead of the correct $\tan{45^\circ}-\frac{\pi}{90} = 0.965$).

Answer 1

The derivative rules that you know for the trig functions are predicated on the angle being in radians. Let $x$ be an angle in radians and $y^\circ=\frac {180}\pi x$ the angle in degrees. We can use the chain rule to write $\frac d{dy} \sin y^\circ=\frac {dx}{dy}\frac d{dx}\sin \frac {\pi x}{180}=\frac \pi{180}\cos (\frac {\pi x}{180})\frac{\pi}{180}=(\frac \pi {180})^2 \cos y^\circ$

Answer 2

$$\tan\left(\frac{\pi}{4}+x\right) = \frac{1+\tan(x)}{1-\tan(x)} $$ and since in a neighbourhood of the origin we have $\tan(x)=x+O(x^3)$, $$ \tan(44^\circ)=\tan\left(\frac{\pi}{4}-\frac{\pi}{180}\right)\approx \frac{180-\pi}{180+\pi}$$ where $\approx$ holds as $\leq$ and the magnitude of the approximation error is $10^{-5}$.

Answer 3

Using radians is a more natural way to express angles. A radian angle of a portion of a circle with arclength $a$ and radius $r$ is defined as $\theta = \frac ar$, from here, to get an angle in degrees you must multiply by a factor of $\frac {180}{\pi}$ and so to compute anything in degrees you have this extra factor floating around. The best way of thinking of this would be, say you have a ruler that measures in centimetres and you want to express the dimensions of a table, you already know that one side of the table is $36$ inches in length and using your ruler you measure the other dimensions to be $80$cm and $75$cm. My best (but probably terrible) analogy of expressing an angle in degrees is like expressing the dimensions of this table as $36$in$\times80$cm$\times0.75$m, technically it is correct but would you really want to use this confusing description? If we wanted to do anything with these numbers, for example compute the volume, we would have some stupid conversion factor floating around and so in the same way that it is more natural to measure area/volume in the same units of length, it is much more natural for angles to be in radians.