Arcsine arcsin(x): Graph, Domain, Range, Derivative and Integral

The arcsine function undoes the sine function: \arcsin x is the angle whose sine is x. Because sine repeats, it is not one-to-one, so arcsine picks the single answer in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]. In calculus it matters mostly as the antiderivative of \frac{1}{\sqrt{1-x^2}}.

Definition of arcsin x

So \arcsin\frac{1}{2} = \frac{\pi}{6}, because \sin\frac{\pi}{6} = \frac{1}{2} and \frac{\pi}{6} is in the allowed range. The notation \sin^{-1}x means the same thing.

Graph of y = arcsin x

The graph is the piece of y = \sin x on \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], reflected across the line y = x.

Key values of arcsine

Properties of arcsine

• Domain [-1, 1], range \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]. Inputs outside [-1, 1] are undefined, since no angle has sine 2.
• Increasing. Arcsine is strictly increasing on its whole domain.
• Odd. \arcsin(-x) = -\arcsin x.
• Cancellation. \sin(\arcsin x) = x for every x in [-1, 1], but \arcsin(\sin x) = x only for x in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right].
• Relation to arccosine. \arcsin x + \arccos x = \frac{\pi}{2}. See arccosine.

Derivative of arcsin x

Why. Let y = \arcsin x, so \sin y = x. Implicit differentiation gives \cos y \cdot y' = 1, so y' = \frac{1}{\cos y}. Since y is in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right], \cos y \ge 0, and \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}. This is a special case of the inverse function rule.

At x = \pm 1 the denominator is zero, which is why the graph has vertical tangents at its endpoints.

Integrals involving arcsine

The derivative formula, read backward, is the integral you will use most:

The integral of arcsine itself comes from integration by parts with u = \arcsin x, dv = dx:

Worked examples

Example 1: chain rule. Differentiate f(x) = \arcsin(3x):

Example 2: a definite integral. \int_0^{1/2}\frac{dx}{\sqrt{1-x^2}} = \arcsin\frac{1}{2} - \arcsin 0 = \frac{\pi}{6}.

Example 3: using a. \int\frac{dx}{\sqrt{9 - x^2}} = \arcsin\frac{x}{3} + C.

Example 4: composition. Simplify \cos(\arcsin x). If \theta = \arcsin x then \sin\theta = x and \cos\theta \ge 0, so \cos(\arcsin x) = \sqrt{1 - x^2}.

Common mistakes

1. Reading \sin^{-1}x as \frac{1}{\sin x}. The reciprocal is \csc x, covered on reciprocal trig functions.
2. Giving an angle outside the range. \arcsin\left(-\frac{1}{2}\right) = -\frac{\pi}{6}, not \frac{7\pi}{6} or \frac{11\pi}{6}.
3. Plugging in |x| > 1. \arcsin 2 does not exist as a real number.
4. Forgetting the chain rule factor. \frac{d}{dx}\arcsin(x^2) = \frac{2x}{\sqrt{1 - x^4}}, not \frac{1}{\sqrt{1-x^4}}.

How arcsine connects to other topics

Arcsine is the backbone of the substitution x = a\sin\theta in trig substitution, used for integrals with \sqrt{a^2 - x^2}. Its siblings are arccosine and arctangent, and all three derivatives are derived the same way in the guide to implicit differentiation.