Graph of arcsin x (inverse sine)

Q: What is the derivative of arcsin x?

1/√(1 − x²), for −1 < x < 1.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The inverse sine graph undoes sine on the interval −π/2 ≤ y ≤ π/2. It runs from (−1, −π/2) to (1, π/2) through the origin, and it is the sine graph reflected in the line y = x.

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The graph of

y=arcsin(x)

for −3 ≤ x ≤ 3. Dots mark its root at the origin.

Key features

Domain: −1 ≤ x ≤ 1
Range: −π/2 ≤ y ≤ π/2
End points: (−1, −π/2) and (1, π/2)
Monotonic: Increasing on its whole domain
Vertical tangents: At x = ±1, where the slope 1/√(1 − x²) is infinite
Symmetry: Odd: arcsin(−x) = −arcsin x
y-intercept: None
Derivative: 1√1−x²
Antiderivative: xarcsin(x)+√1−x²+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of arcsin x looks like

Sine is not one-to-one, so it has no inverse on its whole domain. Restricting it to −π/2 ≤ x ≤ π/2, where it rises from −1 to 1 exactly once, makes it invertible, and arcsin is the inverse of that piece.

Reflecting that piece of the sine curve in the line y = x gives the arcsin graph: a short S-shape that is almost straight near the origin (arcsin x ≈ x for small x) and becomes vertical at its two ends.

The graph only exists for −1 ≤ x ≤ 1, because sine never produces values outside that interval. Asking for arcsin 2 has no real answer.

Points of interest

Found numerically to about twelve digits in the window −3 ≤ x ≤ 3, then shown in exact form where one exists.

x-intercepts (roots): x = 0
Local maxima: None in this window
Local minima: None in this window
Inflection points: (0, 0)

Table of values

A few points to plot, with exact values where they are simple.

Values of arcsin x
x	y = arcsin x
−1	−π/2
−√3/2	−π/3
−1/2	−π/6
0	0
1/2	π/6
√2/2	π/4
1	π/2

How to graph arcsin x by hand

1
Mark the end points (−1, −π/2) and (1, π/2), with π/2 ≈ 1.57, and the origin.
2
Add (1/2, π/6) and (−1/2, −π/6), with π/6 ≈ 0.52.
3
Join them with an S-curve that is nearly straight through the origin and turns vertical at both end points.

Transformations

How the graph moves when you change the formula. The original is arcsin x in blue.

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y = asin(x/2) Domain widened to −2 ≤ x ≤ 2.
y = 2asin(x) Range stretched to −π ≤ y ≤ π.
y = asin(x) + pi/2 Shifted up π/2, so it runs from 0 to π. This equals π − arccos x.