Arcsine Calculator

About Arcsine Calculator

Arcsine is the inverse of sine. It returns the angle whose sine is x, in the range [-90°, 90°].

How to use this calculator

1. Enter a value between \(-1\) and \(1\), because sine values only live in that range.
2. Click calculate to find the principal angle for the arcsine.
3. Read the result in degrees or radians, depending on the calculator setting.
4. Check your work against the sine of the returned angle to confirm it matches the original value.

The formula explained

The formula \(\arcsin(x)=\sin^{-1}(x)\) computes the angle whose sine is \(x\). This gives the principal angle, usually between \(-90^\circ\) and \(90^\circ\), or between \(-\pi/2\) and \(\pi/2\).

• x = the input value, which must be between \(-1\) and \(1\)
• \(\arcsin(x)\) = the angle whose sine is \(x\)

Step by step method

1. Start with the sine value you know, for example \(x=0.5\).
2. Find the angle whose sine equals that number, since \(\sin(30^\circ)=0.5\).
3. Report the principal angle, which is \(30^\circ\) or \(\pi/6\).

Worked example

Problem. Find \(\arcsin(0.5)\).

1. Use the definition of arcsine, so we need the angle whose sine is \(0.5\).
2. Since \(\sin(30^\circ)=0.5\), the principal angle is \(30^\circ\).
3. In radians, \(30^\circ=\pi/6\).

Tips and common mistakes

• Make sure your input is between \(-1\) and \(1\), or arcsine is not defined for real numbers.
• The notation \(\sin^{-1}(x)\) means inverse sine here, not \(1/\sin(x)\).

Frequently asked questions