Free Arcsin Calculator

Understanding the Arcsin Function

The arcsin calculator (often called an inverse sine calculator) is an online trigonometry tool that computes the angle corresponding to a given sine value. By entering a number between –1 and 1 into this sin inverse calculator, you instantly obtain the principal angle in degrees or radians. This makes the inverse trigonometric calculator indispensable for solving right‑triangle problems, verifying manual calculations, and exploring trigonometric relationships.

What Is Arcsin?

Arcsin (arcsine) is the inverse operation of the sine function. For a right‑angled triangle, if the sine of an angle θ equals x, then arcsin(x) returns that angle. Formally:

$$\arcsin(x) = \theta \iff \sin(\theta) = x, \qquad -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}.$$

The restriction on θ guarantees that arcsin is a proper function (one input gives one output). Because the sine of any real number lies in the interval $[ -1, 1 ]$, the arcsine function is defined only for inputs $x$ in that same range. Any attempt to calculate arcsin outside $[-1, 1]$ does not yield a real result.

Principal Value

Sine is periodic, so many angles share the same sine value. For example, $\sin(0) = 0$, but also $\sin(\pi) = 0$, $\sin(2\pi) = 0$, and so on. To provide a unique answer, the arcsine function returns the principal value, which always lies between $-\pi/2$ and $\pi/2$ (or –90° and 90°). This standardized output makes the arcsin function calculator straightforward to use.

Notation

The inverse sine is most commonly written as $\arcsin(x)$ or $\sin^{-1}(x)$. It is important to note that $\sin^{-1}(x)$ does not mean $1/\sin(x)$; the exponent –1 denotes the inverse function, not a reciprocal. In computer programming languages, the same operation is often called asin(x).

Graph of Arcsin x

The graph of $\arcsin(x)$ is obtained by reflecting the graph of $\sin(x)$ restricted to $[-\pi/2, \pi/2]$ across the line $y = x$. The resulting curve has a domain (input values) of $[-1, 1]$ and a range (output angles) of $[-\pi/2, \pi/2]$. The shape is symmetric to the original sine segment, rising from $-\pi/2$ at $x = -1$ to $\pi/2$ at $x = 1$.

Common Arcsin Values

The table below lists frequently encountered sine inputs and their corresponding principal angles:

Sine input ($x$)	Principal angle (degrees)	Principal angle (radians)
$-1$	–90°	$-\dfrac{\pi}{2}$
$-\dfrac{\sqrt{3}}{2}$	–60°	$-\dfrac{\pi}{3}$
$-\dfrac{\sqrt{2}}{2}$	–45°	$-\dfrac{\pi}{4}$
$-\dfrac{1}{2}$	–30°	$-\dfrac{\pi}{6}$
$0$	0°	$0$
$\dfrac{1}{2}$	30°	$\dfrac{\pi}{6}$
$\dfrac{\sqrt{2}}{2}$	45°	$\dfrac{\pi}{4}$
$\dfrac{\sqrt{3}}{2}$	60°	$\dfrac{\pi}{3}$
$1$	90°	$\dfrac{\pi}{2}$

Trigonometric Relationships

Knowing $\arcsin(x)$ lets you derive other trigonometric functions for the same angle. For any valid $x \in [-1, 1]$, the following identities hold:

• $\sin(\arcsin(x)) = x$
• $\cos(\arcsin(x)) = \sqrt{1 - x^{2}}$
• $\tan(\arcsin(x)) = \dfrac{x}{\sqrt{1 - x^{2}}}$

Two additional symmetries are also useful:

$$\arcsin(x) = \frac{\pi}{2} - \arccos(x)$$ $$\arcsin(-x) = -\arcsin(x)$$

Integral and Derivative

When working with calculus, the derivative and integral of the arcsine function are often needed:

• Derivative: $\displaystyle \frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 - x^{2}}}$ for $x \neq \pm 1$
• Integral: $\displaystyle \int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1 - x^{2}} + C$

Using the Arcsin Calculator in Practice

The inverse sine calculator simplifies angle finding. Consider a right triangle where the side opposite angle $\alpha$ has length $a = 6$ and the hypotenuse $c = 10$. The sine of $\alpha$ is $\frac{a}{c} = \frac{6}{10} = 0.6$. Enter $0.6$ (or the fraction $6/10$) into the sin inverse calculator, and it immediately returns the arcsine: the principal angle $\alpha$ is approximately $36.87^\circ$.

This online trigonometry calculator handles any sine input between –1 and 1, delivering the principal angle in both degrees and radians. It is especially valuable for solving triangles, verifying manual work, and exploring relationships between trigonometric functions and their inverses. Applications range from physics and engineering to forensic reconstruction of impact angles.