what does "arc" in arcsin, arccos, arctan stands for

Question

I was just wondering, what does the "arc" in arcsin, arccos, arctan stands for? Is there any particular reason why it is named the way it is?

Answer 1

Each of the "original" functions you're inverting takes an angle, so the inverse returns an angle. But thanks to the arc length formula $s=r\theta$, that's equivalent to specifying an arc length.

Answer 2

These functions are the inverse of their respective trigonometric functions. As such they are multiple valued. Taking the principal value gives the length of the arc on a unit circle subtending an angle that the respective trig function takes as argument.

E.g. $\arccos(0) = \pi/2 +2k\pi$, $k\in\mathbb{Z}$. The principal value is given by taking $k=0$, and we find the arc length of $\pi/2$ on the unit circle subtends the angle of $\pi/2$ having $\cos(\pi/2)=0$.

Answer 3

Let's work with unit circles because they make everything nice.

As we increase $\theta$ starting from $(0,1)$, you increase $y$. We are used to thinking and calculating $\sin \theta = \frac{y}{r}$.

That is, given a particular $\theta$, we have a $y$-coordinate value. Since $r = 1$, $\sin \theta = \frac{y}{r} = y$. Therefore, $\sin \theta$ is the $y$-coordinate of the point on the unit circle after you have traveled an angular distance of $\theta$.

In addition, the arc length you have travelled is given by $s = r \theta = \theta$. We may simply recast our previous statement as: $\sin \theta$ is the $y$-coordinate of the point on the unit circle after you have traveled an arc length of $\theta$.

With this setup, we may ask the inverse question: "given a specific $y$ coordinate, what arc length was needed in order to arrive there?". That arc length, as we discussed above for a unit circle, is the same as the angle. Essentially, this is the ``arc" in the inverse trigonometric functions comes from.

We can read $\theta = \arcsin(y)$ as: "what is the arclength I have travelled along the unit circle to get to $y$?". Once you know the arc length, you also know the angle since they are the same value.