The arcsin calculator answers one question: which angle has this sine? Enter a value between −1 and 1 and it returns the principal value in degrees and radians, the exact form where one exists, and — importantly — the other angle with the same sine, which is the one most calculators never mention.
Arb Digital publishes this alongside separate pages for arccos and arctan, because the three inverse functions are not interchangeable. Each has a different output range, and that range is the whole story of what an inverse trigonometric function can and cannot tell you. Arcsin's range is [−π⁄2, π⁄2], or −90° to 90° inclusive. Everything distinctive about arcsin follows from that one fact.
What This Arcsin Calculator Does
Give it a sine value and it returns the principal arcsine — the single angle in [−90°, 90°] whose sine matches your input. Alongside it you get the same angle in the other unit, so you never have to convert, and the supplementary angle 180° minus the result, which shares the identical sine and is a legitimate answer to "what angle has this sine?" in most real problems.
At the standard values it also prints the exact form. arcsin(0.5) is not merely 30° — it is exactly π⁄6. arcsin(√3⁄2) is exactly π⁄3. If you type a value outside [−1, 1], the calculator does not return NaN or an error code. It explains that sine's output is bounded by the unit circle's radius, so no real angle can have a sine of 1.4, and tells you what to check.
How to Use It
- Enter the sine value. This is usually a ratio you already computed — opposite over hypotenuse in a right triangle, or a value from a physics equation.
- Choose degrees or radians. Degrees is the default; radians is what you want if the angle feeds into calculus or a programming language.
- Set decimal places to whatever precision your work needs.
- Read the principal value, which is the direct answer and always sits between −90° and 90°.
- Check the second angle before you finish. If your problem allows an obtuse angle, that supplement may be the answer you actually want.
The Formula: How It's Calculated
Arcsine is defined as the inverse of sine, restricted so that an inverse can exist at all. Sine on its own is not invertible: it takes the value 0.5 at 30°, at 150°, at 390°, at 510°, and at infinitely many other angles. A function cannot have more than one output, so mathematics chooses one branch by convention — the stretch of the sine curve from −90° to 90°, where sine climbs steadily from −1 to 1 without ever repeating a value.
So y = arcsin(x) means sin y = x with y restricted to [−π⁄2, π⁄2]. The domain is [−1, 1] and the range is [−π⁄2, π⁄2]. Working through the default input: arcsin(0.5) asks which angle in that band has sine 0.5. The answer is π⁄6 radians, or 30°, because the point on the unit circle 30° anticlockwise from the x‑axis has height exactly one half. A negative input mirrors it: arcsin(−0.5) = −30°, since arcsine is an odd function — arcsin(−x) = −arcsin(x). The Wolfram MathWorld entry on the inverse sine sets out the branch convention formally.
Why arcsin(sin 150°) Is 30°, Not 150°
This is the practical consequence of the restricted range, and it catches almost everyone the first time. Take 150°. Its sine is 0.5. Feed 0.5 back into arcsine and you get 30°. The angle you started with is gone, and no amount of care with the arithmetic will bring it back — the information was destroyed the moment you took the sine, because sine maps both 30° and 150° to the same number.
The round trip only works for angles already inside [−90°, 90°]. Outside that band, arcsin(sin θ) returns a different angle with the same sine, not θ itself. This matters wherever an angle is recovered from a measured ratio. If you calculate an angle of elevation as arcsin(opposite ÷ hypotenuse) you are safe, because an elevation is acute by construction. But if you are solving a triangle where an angle might be obtuse — the SSA case handled on our law of sines calculator — arcsine will hand you the acute answer every time, and the obtuse triangle disappears unless you deliberately check the supplement. That is exactly why the ambiguous case exists, and why this page shows 180° minus the result as a first-class output.
Arcsin, Arccos and Arctan Are Not Interchangeable
All three inverse functions restrict a branch, but they restrict different ones, and swapping between them changes which answers are reachable.
Arcsin covers [−90°, 90°] — the right half of the unit circle, quadrants I and IV. It can return negative angles but never obtuse ones. Arccos covers [0°, 180°] — the top half, quadrants I and II. It can return obtuse angles but never negative ones, which is why the arccos calculator is the right tool for triangle interior angles and the law of cosines never suffers ambiguity. Arctan covers (−90°, 90°) open at both ends and, uniquely, accepts every real number as input, which the arctan calculator explains in full.
A rule that avoids most trouble: if the angle you are recovering could be obtuse, do not use arcsine. Restructure the problem so you can use arccosine instead, typically by finding a side with the law of cosines first. Arcsine is the right choice when the angle is known to be acute or when a signed angle around zero is what you want.
Why Inputs Outside −1 to 1 Have No Answer
Sine is the vertical coordinate of a point on a circle of radius 1. A point on that circle cannot sit higher than 1 or lower than −1, so sine's output is permanently trapped in [−1, 1]. Asking for arcsin(1.4) is asking which angle has a height of 1.4 on a circle whose top is at 1. No such angle exists among the real numbers.
When that happens in practice, it is almost never a request for exotic mathematics — it is a signal that something upstream is wrong. The three usual causes: you divided the wrong way round, putting the hypotenuse on top when the hypotenuse must always be the larger of the two; you used a leg instead of the hypotenuse as the denominator; or the measurements themselves are inconsistent, as when three surveyed lengths do not actually form a triangle. Check the ratio's construction first. Our Pythagorean theorem calculator is a quick way to confirm which side really is the hypotenuse before you take any ratio.
Exact Values Worth Recognising
Six inputs produce clean exact answers, and knowing them by sight saves time and prevents rounding damage: arcsin(0) = 0, arcsin(1⁄2) = π⁄6 (30°), arcsin(√2⁄2) = π⁄4 (45°), arcsin(√3⁄2) = π⁄3 (60°), and arcsin(1) = π⁄2 (90°). Negative inputs give the negatives of those same angles.
The decimal traps are worth naming. √2⁄2 is 0.7071067812, and typing 0.707 returns 44.991° rather than 45°. √3⁄2 is 0.8660254038; typing 0.866 gives 59.997°. Neither is a calculator fault — you asked a slightly different question. If a result comes out a hundredth of a degree off a round number, suspect the input's precision before anything else. The trigonometric functions calculator will give you the full-precision sine of any standard angle to feed back in, and the angle converter handles conversion if you need gradians as well.
Where Arcsine Turns Up in Practice
The classic application is any right-triangle problem where the opposite side and the hypotenuse are the measured quantities: a ramp of known length rising a known height, a ladder against a wall, a cable of known length spanning a known vertical drop. In each case the angle is arcsin(rise ÷ length), and the acute-only range is exactly right because these angles are acute by construction.
In optics, Snell's law is solved with arcsine to find a refraction angle, and the out-of-range condition has a real physical meaning there — an input above 1 signals total internal reflection, where no refracted ray exists at all. In signal processing, arcsine recovers a phase angle from a normalised amplitude. In each of these, the restricted range is a feature, not a limitation: the situation genuinely has an acute answer. The NIST Digital Library of Mathematical Functions gives the formal definitions and branch cuts for all the inverse trigonometric functions.
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Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Assuming arcsin(sin θ) returns θ — it only does so for angles already between −90° and 90°. Outside that band you get a different angle with the same sine.
- Missing the obtuse solution — if your problem permits an angle above 90°, check 180° minus the principal value before concluding.
- Entering a ratio bigger than 1 — usually the hypotenuse ended up in the numerator. Sine's output cannot exceed the circle's radius.
- Typing 0.707 instead of the full √2⁄2 — a truncated input gives 44.99° rather than 45°, and it is your rounding, not the calculator's.
- Reading the output in the wrong unit — 0.5236 radians and 30° are the same angle; confusing them is the most common inverse-trig error of all.
Related Free Tools From Arb Digital
Compare with the arccos calculator when the angle might be obtuse, and the arctan calculator when you have a ratio of two legs rather than a leg and the hypotenuse. Go forwards instead of backwards with the trigonometric functions calculator, see the geometry on the unit circle calculator, solve whole triangles with the law of sines calculator, or browse the full free online tools hub.
Frequently Asked Questions
[−π⁄2, π⁄2] in radians, or −90° to 90° in degrees, inclusive at both ends. Arcsine never returns an obtuse angle, which is the single most important thing to know about it.
Sine is the height of a point on a circle of radius 1, so its value can never exceed 1 or fall below −1. No real angle has a sine of 2. An out-of-range input usually means a ratio was formed upside down, with the hypotenuse in the numerator.
Because 30° and 150° have the same sine, and arcsine must return exactly one angle. It always chooses the one in its restricted range, so the original 150° is not recoverable from the sine value alone.
Nothing — they are two notations for the same function. The superscript −1 here means inverse function, not a reciprocal. The reciprocal of sine is cosecant, which is a different thing entirely.
π⁄6 radians, which is 30°. The exact form is preferable to the decimal 0.5236 whenever the value will be used in further algebra.
Whenever the angle you are finding could be obtuse. Arccosine's range covers 0° to 180°, the full span of a triangle's interior angle, so it returns the correct angle without any supplement check.
Yes. arcsin(−x) equals −arcsin(x), so a negative input simply mirrors the result below zero. This is why arcsine can return negative angles while arccosine cannot.
This tool is provided for education and reference. Verify any result used in engineering, optics or navigation work against an independent calculation.
