tan^-1
|ark-tan|
🇺🇸
/ˈɑrk.tæn/
🇬🇧
/ˈɑːk.tæn/
angle whose tangent equals a given value
Etymology
'tan^-1' as a notation corresponds to the spoken term 'arctan', which originates from the phrase 'arc tangent' in English; 'tangent' comes via Latin 'tangens' (present participle of 'tangere') meaning 'touching'.
'tangent' entered mathematical vocabulary from Latin 'tangens' (from 'tangere', 'to touch') via medieval Latin and Old French; the phrase 'arc tangent' arose in English to mean the arc (angle) corresponding to a given tangent value; this was contracted in usage to 'arctan', and the compact notation tan^-1 emerged in the 18th–19th centuries alongside notation for inverse trigonometric functions.
Initially 'tangent' referred to a line 'touching' a circle; over time 'tangent' named the trigonometric function, and 'arc tangent' came to mean the angle whose tangent has a given value; the modern meaning as the inverse function (arctan, tan^-1) is consistent with that.
Meanings by Part of Speech
Noun 1
the inverse function of the tangent function: for a real number y, tan^-1(y) (also written arctan(y) or atan(y)) is the angle x whose tangent is y. Domain: all real numbers. Principal value (common convention): range is (-π/2, π/2).
tan^-1(1) = π/4
Synonyms
Antonyms
Noun 2
notation warning / alternate usage: some texts or calculators use tan^-1 to mean the multiplicative inverse 1/tan (i.e., cotangent). This is nonstandard in most higher mathematics; tan^-1 is normally the inverse function (arctan).
Be careful: some old calculators display tan^-1 for cot x, but most mathematical texts use tan^-1 for arctan x.
Synonyms
Last updated: 2026/01/14 19:09