What does arcsine mean?
Arcsine is the inverse of the trigonometric sine function. Where sine takes an angle and returns a ratio between −1 and 1, arcsine performs the opposite operation: given such a ratio, it returns the angle whose sine equals it. Because sine is not one-to-one over all angles, mathematicians restrict arcsine to its principal values, which lie between −π/2 and π/2 radians (−90° to 90°); this ensures every input yields exactly one output. The function is standardly abbreviated arcsin or written sin⁻¹, and appears routinely in calculus, geometry, physics, and signal processing. It also has a notable statistical application: the arcsine transformation converts proportion data into a form better suited to analysis of variance. Though purely technical in register, arcsine is fundamental to any work that requires recovering an angle from a known ratio, making it one of the essential inverse trigonometric functions alongside arccosine and arctangent.
nounThe inverse of the sine function: for a given value between −1 and 1, the angle whose sine is that value. It is typically written as arcsin or sin⁻¹ and returns an angle, usually expressed in radians.
- The inverse trigonometric function that returns the angle whose sine is a given number, restricted to the range −π/2 to π/2 radians (the principal value).
"To recover the angle from its sine value of 0.5, take the arcsine, which gives π/6 radians."
"Since sin(θ) = 0.5, we find θ = arcsine(0.5) = π/6."
"In statistics, the arcsine transformation stabilizes the variance of proportions before analysis."
Rarely pluralized in practice; the plural may appear when comparing multiple branches or instances of the function across contexts.
"Different branches of the multivalued relation give several distinct arcsines for the same input."
The 'arc' in arcsine isn't about curves at all — it's there because early mathematicians described trigonometric values as lengths of arcs on a circle.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of arcsine
The word arcsine combines 'arc', from the Latin arcus meaning bow or curve, with 'sine', which descends from the Latin sinus. The term reflects an older way of thinking about trigonometry in which function values were understood as the lengths of arcs on a circle rather than as abstract ratios. It emerged in mathematical literature as inverse trigonometric functions were formalized, generally attributed to developments in the seventeenth and eighteenth centuries. Its sibling terms arccosine and arctangent follow the same pattern, all built on the Latin-derived names of their parent functions.
How arcsine is actually used
A technical term used almost exclusively in mathematics, physics, engineering, and statistics; it would rarely appear in general conversation. In writing it is usually abbreviated as arcsin or denoted sin⁻¹, though the latter notation can be misread as the reciprocal 1/sin. The function is multivalued unless restricted to its principal branch, so calculators return only values between −90° and 90° (−π/2 and π/2 radians).
Easily confused with arcsine
Arsine is a highly toxic chemical compound of arsenic and hydrogen (AsH₃), whereas arcsine is a mathematical function; the two differ by only one letter.
Sine maps an angle to a ratio between −1 and 1, while arcsine does the reverse, mapping such a ratio back to an angle.