Calculate arcsin (sin⁻¹) of any value from −1 to 1. Shows result in degrees, radians, gradians, general solution, domain check, and common values reference.
The **Inverse Sine Calculator** computes arcsin(x) — the angle whose sine equals x — and presents the result simultaneously in degrees, radians, and gradians. It performs automatic domain validation (arcsin is defined only for inputs between −1 and 1), making it impossible to get a confusing NaN result without explanation.
The inverse sine function, written sin⁻¹(x) or arcsin(x), answers the question "what angle has a sine value of x?" Its principal value lies in the range −90° to +90° (−π/2 to π/2 radians). Because sine is periodic, every valid input actually corresponds to infinitely many angles; this calculator displays five terms of the general solution θ = nπ + (−1)ⁿ · arcsin(x) so you can see the pattern.
Beyond the primary result, the tool shows related inverse trig values (arccos and arctan of the same input), verifies the complementary identity arcsin(x) + arccos(x) = 90°, and includes visual range and domain indicators so you can instantly see where your result falls. A reference table lists exact arcsin values for nine standard inputs (−1, −√3/2, −√2/2, −1/2, 0, 1/2, √2/2, √3/2, 1), and a collapsible range generator lets you compute arcsin for any sequence of values with a custom step size. Eight preset buttons cover the most commonly needed inputs, and a collapsible identities panel summarizes key properties including the derivative, integral, and Taylor series expansion.
Inverse Sine Calculator (arcsin / sin⁻¹) helps you avoid repetitive setup mistakes when solving trigonometric and coordinate-geometry problems. Instead of recalculating conversions, signs, and edge cases by hand, you can test inputs immediately, inspect intermediate values, and confirm final answers before submitting work or using numbers in downstream calculations. It surfaces key outputs like sin⁻¹(x), Degrees, Radians in one pass.
arcsin(x) returns the angle θ in [−90°, 90°] such that sin(θ) = x. General solution: θ = nπ + (−1)^n · arcsin(x), n ∈ ℤ. Complement: arcsin(x) + arccos(x) = π/2.
Result: 30°
Using x=0.5, the calculator returns 30°. This example mirrors the calculator's live computation flow and is useful for checking manual steps and unit handling.
This calculator is tailored to inverse sine calculator (arcsin / sin⁻¹) workflows, including common input modes, unit handling, and special-case behavior. It is designed for fast checking during homework, exam preparation, technical drafting, and coding tasks where trigonometric consistency matters.
Use the primary result together with supporting outputs to verify direction, magnitude, and validity. Cross-check against known identities or geometric constraints, and confirm that angle ranges, sign conventions, and domain restrictions are satisfied before using the numbers elsewhere.
A reliable way to improve is to solve once manually, then verify with the calculator and explain any mismatch. Repeat this on varied examples and edge cases. The built-in preset scenarios for quick trials, comparison tables for side-by-side validation, visual cues that make trends and quadrants easier to read help you build pattern recognition and reduce sign or conversion errors over time.
Arcsin, written sin⁻¹(x) or arcsin(x), is the inverse of the sine function. It returns the angle whose sine is x. The principal value is in [−90°, 90°].
The domain is [−1, 1]. Since sine only produces values between −1 and 1, arcsin is undefined for any input outside this range.
The principal-value range is [−90°, 90°] or [−π/2, π/2] in radians. This restriction makes arcsin a true function (one output per input).
θ = nπ + (−1)^n · arcsin(x), where n is any integer. This generates all angles with the given sine value: the principal value, its supplement, and their periodic repetitions.
arcsin(x) + arccos(x) = π/2 (90°) for every x in [−1, 1]. If arcsin(0.5) = 30°, then arccos(0.5) = 60°.
No. sin⁻¹(x) means the inverse sine (arcsin), NOT the reciprocal. The reciprocal 1/sin(x) is called cosecant (csc x). This notation is a common source of confusion.