How this instrument works
Arcsin answers a single question: given a value x between −1 and 1, which angle has that number as its sine? The name is literal — 'arc' plus 'sine' meant, in the eighteenth-century notation of Bernoulli and Euler, the arc of the unit circle whose sine equals x, since on a circle of radius one an arc length and its central angle in radians are the same number. Writing sin⁻¹ for this and then reading it as the reciprocal cosecant is the slip that name was meant to head off.
The domain stops at ±1 because sine itself never leaves that band, and the range is squeezed to −90° through 90° for a sharper reason: sine repeats every full turn and is not one-to-one, so sin(30°) and sin(150°) both equal 0.5. Mathematicians fix this by choosing the one stretch where sine climbs steadily from −1 to 1 without doubling back — the principal branch — and arcsin always reports the angle from that stretch, never its mirror twin further round the circle.
Differentiate arcsin(x) and you get 1 ⁄ √(1 − x²), a slope that stays modest near x = 0 and shoots toward infinity as x approaches ±1. Geometrically the sine curve is nearly flat at its peak, so a hair's difference in x near the edge of the domain swings the recovered angle far more than the same difference does near the middle — a steepening that catches anyone who expects the inverse to behave as evenly as the original.
- Enter your sine value in the x field — any figure from −1 to 1 is accepted, stepping by 0.05 if you drag it.
- The arcsin(x) field returns the principal angle, shown in degrees by default.
- Switch its unit dropdown to rad for radians or turn for fractions of a full circle.
- Try x = 1 or x = −1 to see the calculator return exactly 90° or −90°, the two edges of arcsin's range.
Worked example — recovering 30° from x = 0.5
Set x to 0.5 — say, a brace whose vertical rise is exactly half the length of its diagonal strut, since sin(θ) equals rise divided by strut length. The arcsin(x) field returns 0.5235987755982989 in radians, or 30° once the unit dropdown is flipped to degrees, because sin(30°) equals 0.5 to the last digit.
That radian figure is not a rounded stand-in; it is π ⁄ 6, the exact measure of a 30° angle, carried to the precision the calculator keeps internally. Push x to 0.6 instead and the returned angle grows to roughly 36.87°, not 36° — a reminder that arcsin is not a straight-line function, only close to one near the middle of its domain.
Questions
What is the range of arcsin, and why is it limited to −90° through 90°?
Sine repeats every 360° and gives the same value for infinitely many angles — sin(30°) and sin(150°) are both 0.5 — so an inverse function needs one angle per input, not several. Arcsin's output is fixed to the stretch from −90° to 90°, where sine climbs steadily without repeating, known as the principal range.
How does arcsin differ from arccos, given they share the same domain?
Both accept x from −1 to 1, but their ranges sit in different places: arcsin reports an angle from −90° to 90°, straddling zero, while arccos reports one from 0° to 180°, never negative. The two are linked by arcsin(x) + arccos(x) = 90° for any valid x, since sine and cosine are complementary angles.
Why does arcsin have a restricted domain when arctan does not?
Arcsin inverts sine, and sine only ever produces values between −1 and 1, so anything outside that band has no angle to return, and the calculator rejects it. Tangent, by contrast, takes on every real number as an angle approaches ±90°, so arctan accepts any real x with no boundary to enforce.
What is the most common mistake people make with arcsin?
Assuming arcsin(sin(x)) always returns x. It only does inside the principal range, −90° to 90°. Feed it an angle outside that band, such as 150°, and arcsin(sin(150°)) comes back as 30° — the principal angle that shares the same sine, not the original 150°.
How do I find the derivative of arcsin without memorising it?
Start from sin(θ) = x and differentiate both sides: cos(θ) dθ/dx = 1, so dθ/dx = 1 ⁄ cos(θ). Since cos(θ) = √(1 − sin²θ) = √(1 − x²) inside the principal range, where cosine stays positive, that gives d/dx arcsin(x) = 1 ⁄ √(1 − x²) directly from the definition, with no table to recall.
Why does the calculator reject an x value like 1.2?
Because 1.2 sits outside sine's own output range — no real angle has a sine greater than 1 or less than −1. The check exists so the sheet never silently returns a wrong or complex result; it flags the input instead of guessing at one.