Derivative of the $\sin(x)$ when $x$ is measured in degrees

Question

So a classic thing to derive in calculus textbooks is something like a statement as follows

Is $\frac{d}{dx}\sin(u)$ the same as the derivative of $\frac{d}{dx}\sin(x)$ where $u$ is an angle measured in degrees and $x$ is measured in radians? and of course the answer is no because of the chain rule.

Except usually this is ambiguously worded as "Is the derivative of $\sin(u)$, where $u$ is measured in degrees, equal to the derivative of $\sin(x)$ where $x$ is the same angle but measured in in radians?"

Then the texts go on to say something like "No and this why we don't work in degrees and instead chose to work in radians, to avoid all the messy constants that come out of taking derivatives." Am I crazy by thinking this is an odd thing to say that will end up confusing students. If your independent variable was an angle measured in degrees, you are probably more interested in it's derivative with respect to degrees not radians, which would infact be equal at the corresponding degrees and radians of an angle. Is my understanding wrong here. Is what the books say fine? I think at minimum they should at least be clear that we are taking the derivative with respect to radians, no?

Note this is not a duplicate of

Derivative of the sine function when the argument is measured in degrees

Even though it is highly related.

Answer 1

The essential problem is writing "$\sin u$, where $u$ is measured in degrees". This doesn't really mean anything. If you want degrees, you need to write them explicitly: $\sin u^\circ$. If you want the sine of an angle, you need to write that: $\sin\angle ABC$ or similar. At some point in geometry or trigonometry class, someone should be teaching that $\vphantom t^\circ=\frac{2\pi}{360}$.

Answer 2

Whether an angle is measured in degrees or radians is most definitely worth bringing up in a class/lecture so the students can get a solid understanding of derivatives.

UNITS DO MATTER!!

Of course they do. Saying units do not matter is like claiming that $50$ miles per hour is equivalent to 50 feet per hour or 50 miles per second.

For x measured in radians, $d/dx(\sin x) = \cos x$

For x measured in degrees, $d/dx(\sin x) = (\pi/180) \cos x$

A derivative is the slope of a line, the change in the vertical units per change in the horizontal units. When you go from radians to degrees, the change in vertical units remains constant while the change in horizontal units increases tremendously (which, in turn, has the derivative become a lot smaller).

A good way to see this is simply to graph $\sin x$ in degrees. If the derivatives are all the same as they are when measured in radians you should find a tangent line with slope $1$ somewhere (as is true for $\sin x$ at $x=0$ radians). You will not find any tangents with slopes nearly this large when your angles are measured in degrees.

What you guys are claiming is "obvious" is patently false.

Answer 3

Yes, you're describing the same object $x$, but units matter. That is, the units you use to measure that object matter in describing what you mean. Like, 1 meter and 3.2808 feet are practically the same length, but are described through different units of measure.

And in Calculus, radians is a unit of measure that will give you the least amount of headaches (less tracking of constants, etc.) when calculating derivatives, integrals, etc. It's similar to why $\log_e=\ln$, in Calculus, is called the natural logarithm: $\frac{d}{dx}\ln x=\frac{1}{x}$, while $\frac{d}{dx}\log_a x=\frac{1}{x\ln a}$.

In fact, radians (as a measure of angles) is arguably a better measurement to define and use than degrees. Given a circle of radius $r$, if you're interested in the length of the arc of a circle, arc given by an angle $\theta$:

In degrees: the length of the arc of a circle is $r\left(\frac{\theta\pi}{180}\right)$

In radians: the length of the arc of a circle is simply $r\theta$.

Fundamentally, calculating $\frac{d}{dx}\sin x$ boils down to solving the limit $\lim_{x\to 0}\frac{\sin x}{x}$. And what does one mean by $\frac{\sin x}{x}$?

Geometrically, and seeing $x$ in radians, one can say $\frac{\sin x}{x}$ is simply the ratio of the length of a side of a right-triangle ($\sin x$) and the length of an arc of a circle ($x$). Describing $\frac{\sin x}{x}$ this way, we get a ratio of lengths.