Inverse Sine
TrigonometryInverse 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.
Example
If $\sin(\theta) = 0.5$, then $\theta = \arcsin(0.5) = 30^\circ$. You can use the $\sin^{-1}$ button on a calculator.
Key Insight
Think of it as "undoing" the sine. Sine takes an angle and gives a number. Inverse sine takes a number and gives back the angle.
Definition
$\arcsin(x)$ is the inverse function of sin restricted to $[-\pi/2, \pi/2]$. For any $x$ in $[-1, 1]$, $\arcsin(x)$ is the unique angle $\theta$ in $[-\pi/2, \pi/2]$ such that $\sin(\theta) = x$. The range is always between $-90^\circ$ and $90^\circ$.
Example
A right triangle has opposite $= 7$, hypotenuse $= 10$. $\sin(\theta) = 7/10 = 0.7$, so $\theta = \arcsin(0.7) \approx 44.4^\circ$.
Key Insight
The restriction to $[-\pi/2, \pi/2]$ is necessary because sine is not one-to-one over all angles. The same sine value (e.g., $0.5$) corresponds to infinitely many angles ($30^\circ$, $150^\circ$, $390^\circ$, $\ldots$), so we restrict the domain to make inverse sine a true function.
Definition
$\arcsin: [-1, 1] \to [-\pi/2, \pi/2]$ is a continuous, strictly increasing function. Its derivative is $d/dx[\arcsin(x)] = 1/\sqrt{1 - x^2}$ for $|x| < 1$. The power series is $\arcsin(x) = x + x^3/6 + 3x^5/40 + 15x^7/336 + \ldots$ for $|x| < 1$.
Example
$\arcsin(x) = \int_0^x dt/\sqrt{1 - t^2}$. This integral form is fundamental in computing arc length of ellipses and in probability (arcsine distribution in random walks).
Key Insight
The arcsine distribution $P(X \le x) = (2/\pi)\arcsin(\sqrt{x})$ on $[0,1]$ appears in the arc-sine law of Brownian motion: the fraction of time a random walk spends positive follows this distribution, a deep result in probability theory.