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

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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?

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2 Answers

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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.

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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 $\cos(\pi/2)=0$.

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