Inverse Sine
Inverse sine (arcsin) is the function that returns the angle whose sine equals a given value.
Formula
\arcsin(x) = \theta \text{ such that } \sin(\theta) = x, \text{ for } x \in [-1, 1]
Definition
Inverse sine (also called arcsin or $\sin^{-1}$) is the reverse of sine: if sine tells you the ratio given the angle, inverse sine tells you the angle given the ratio. Formally, $\arcsin(x)$ is the inverse of sin restricted to $[-\pi/2, \pi/2]$, so for any $x$ in $[-1, 1]$, $\arcsin(x)$ is the unique angle $\theta$ in that range with $\sin(\theta) = x$; this restriction is necessary because the same sine value corresponds to infinitely many angles ($30^\circ$, $150^\circ$, $390^\circ$, $\ldots$), so the domain must be restricted to make inverse sine a true function. It is continuous and strictly increasing, with derivative $d/dx[\arcsin(x)] = 1/\sqrt{1 - x^2}$ for $|x| < 1$.
Example
If $\sin(\theta) = 0.5$, then $\theta = \arcsin(0.5) = 30^\circ$, found with the $\sin^{-1}$ button on a calculator; for a right triangle with opposite $= 7$ and hypotenuse $= 10$, $\sin(\theta) = 0.7$, so $\theta = \arcsin(0.7) \approx 44.4^\circ$. Written as an integral, $\arcsin(x) = \int_0^x dt/\sqrt{1 - t^2}$, a form fundamental in computing the arc length of ellipses and in probability, where the arcsine distribution $P(X \le x) = (2/\pi)\arcsin(\sqrt{x})$ on $[0,1]$ appears in the arc-sine law of Brownian motion.
Key Insight
Think of it as "undoing" sine: sine takes an angle and gives a number, inverse sine takes a number and gives back the angle in $[-90^\circ, 90^\circ]$. The arc-sine law describes the fraction of time a random walk spends positive, a deep result in probability theory that traces directly back to this restricted inverse function.