Arcsin Calculator

What this calculator does

So, you know the sine of an angle and you want the angle back. This tool does that reverse step: you give it a sine value, and it returns the angle, shown both in radians and in degrees to three decimal places.

There is a single input box for the sine value. This is the sine calculator run backwards: that one turns an angle into a sine, this one turns a sine into an angle.

How to use it

1. Type the sine value, a number between -1 and 1.
2. Press Calculate.

The tool gives the answer two ways at once, in radians and in degrees, so you can take whichever your work needs without converting.

Arcsine undoes sine

Sine takes an angle and gives a ratio. Arcsine, written arcsin or sometimes sin with a small -1, goes the other way: it takes that ratio and gives back the angle. The "arc" in the name is the clue to what it means. It stands for "the arc, or angle, whose sine is this number." So arcsin(0.5) is read as "the angle whose sine is 0.5", which is 30 degrees. Whenever you have measured or been given a sine and need the angle behind it, arcsine is the operation that recovers it.

Why the input must be between -1 and 1

The tool only accepts a value from -1 to 1, and refuses anything outside that band. This is not a limitation of the tool, it is a fact about sine. The sine of any angle is the height of a point on a circle of radius 1, so it can never be more than 1 or less than -1. That means no angle anywhere has a sine of 2, or of -1.5. Asking for arcsin(2) is asking for an angle that does not exist, so the tool stops and tells you to enter a value in the valid range.

Why it gives one answer, not many

Here is the subtle part of every inverse trig function. Sine repeats, so many different angles share the same sine. The sine of 30 degrees is 0.5, but so is the sine of 150 degrees, and of 390 degrees, and infinitely many others. If arcsine tried to return all of them, it would never be a proper function. So it returns just one, the standard choice, called the principal value, which for arcsine lies between -90 and 90 degrees. The tool gives you that principal angle. If your situation calls for a different one of the matching angles, you work it out from the principal value using the symmetry of the sine wave.

A worked example

Enter 0.5. The tool asks, in effect, which angle has a sine of 0.5, and returns 30 degrees, or about 0.524 radians. Enter 1, the largest allowed value, and it returns 90 degrees, the top of the circle. Enter 0 and it returns 0 degrees. Enter something like 1.5 and it declines, because no angle has a sine that large.

Radians and degrees

The answer comes in both units so you do not have to choose in advance. Degrees are the everyday measure, a right angle being 90. Radians measure the angle by arc length on the unit circle, a right angle being pi over 2, and they are the unit most higher mathematics runs on. For arcsine, the degree answer always falls between -90 and 90, and the radian answer between minus pi over 2 and pi over 2.

Questions people ask

What does arcsine do?

It takes a sine value and returns the angle that has that sine. It is the inverse of the sine function.

Why won't it accept a number bigger than 1?

Because sine never exceeds 1 or drops below -1, so no angle has a sine outside that range. Such an input would ask for an angle that does not exist.

Why only one answer when many angles fit?

Sine repeats, so infinitely many angles share a sine value. Arcsine returns the single standard one, the principal value between -90 and 90 degrees.

Which unit is the answer in?

Both. It shows the angle in radians and in degrees at the same time, so you can use whichever you need.

How does this relate to the sine calculator?

They are opposites. The sine calculator turns an angle into a sine value; this one turns a sine value back into an angle.

References

On the inverse sine and its principal range. Arcsine is the inverse of the sine, defined for inputs from -1 to 1, and it returns a single principal angle rather than all the angles that share a sine.

1. Eric W. Weisstein, "Inverse Sine," from MathWorld, a Wolfram resource, on the inverse function of the sine and its principal values.
2. Eric W. Weisstein, "Inverse Trigonometric Functions," from MathWorld, a Wolfram resource, on the inverse trigonometric functions and the conventions for their ranges.