Inverse Sine (Arcsine) $y = \arcsin x$

$\arcsin x$ (the inverse sine, or arcsine) is the inverse of the sine function $\sin$. Because $\sin$ is periodic, it is not one-to-one over its whole domain. We therefore restrict it to the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ on which it is increasing and define the inverse there (the principal value) as $\arcsin$.

Definition

$y = \arcsin x$ is the value $y$ satisfying $\sin y = x$ with $-\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2}$.

Domain and range

The domain is $-1 \le x \le 1$, that is $[-1, 1]$, because $\sin$ only takes values between $-1$ and $1$. The range is $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$.

Symmetry and monotonicity

$\arcsin$ is an odd function with $\arcsin(-x) = -\arcsin x$, so its graph is symmetric about the origin. Its derivative

is always positive, so the function is strictly increasing over its whole domain.

Notable points and tangents

The graph passes through the three points $\left(-1, -\dfrac{\pi}{2}\right)$, $(0, 0)$, and $\left(1, \dfrac{\pi}{2}\right)$. At the endpoints $x = \pm 1$ the denominator of the derivative tends to $0$, so the slope becomes infinite and the tangent is vertical. Near the origin $\arcsin x \approx x$.

Relation to other functions

There is an important relation with the inverse cosine:

so their sum is always a right angle.

Specific values

For example $\arcsin \dfrac{1}{2} = \dfrac{\pi}{6}$, $\arcsin \dfrac{\sqrt{2}}{2} = \dfrac{\pi}{4}$, and $\arcsin 1 = \dfrac{\pi}{2}$.

Applications

It is widely used for inverse problems that recover an angle, for phase calculations in simple harmonic motion and waves, for solving triangles, and for evaluating the integral $\displaystyle\int \frac{dx}{\sqrt{1 - x^2}}$.