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

$\arcsin x$, the inverse sine or arcsine, is the inverse of the sine function $\sin$¹. Because $\sin$ is periodic it is not one-to-one over its whole domain, so it is restricted to the interval $\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]$, on which it increases, and the inverse of that restriction, the principal value, is $\arcsin$.

Definition

$y = \arcsin x$ is the value $y$ satisfying both of the following.

• $\sin y = x$
• $-\dfrac{\pi}{2} \leq y \leq \dfrac{\pi}{2}$

Domain and range

The domain is the interval $[-1, 1]$, since $\sin$ takes no values outside $-1$ to $1$. The range is the interval $\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]$.

Symmetry and monotonicity

Since $\arcsin(-x) = -\arcsin x$ it is an odd function, and its graph is symmetric about the origin. The derivative is as follows.

It is always positive, so the function increases monotonically over its whole domain.

Notable values

Tangents at the endpoints

At $x = \pm 1$ the denominator of the derivative tends to $0$, so the slope grows without bound and the tangent becomes vertical. The horizontal tangents that $\sin$ has at $x = \pm\dfrac{\pi}{2}$ appear as vertical tangents on the inverse. Near the origin $\arcsin x \approx x$.

Relation to the inverse cosine

The two always add to a right angle: whatever $\arcsin$ gains, $\arccos$ gives up, so the sum never moves.

Series expansion

The expansion about the origin converges on the interval $[-1, 1]$.

Applications

• Inverse problems that recover an angle from the value of its sine
• Phase calculations in simple harmonic motion and waves
• Solving triangles, where the law of sines gives an angle
• The integral $\int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C$

1. Inverse trigonometric functions, Wikipedia