arcsec: meaning, definition, pronunciation and examples
C2Formal Technical/Mathematical
Quick answer
What does “arcsec” mean?
The inverse of the secant function (sec⁻¹(x)), an inverse trigonometric function that returns the angle whose secant is a given number.
Audio
Pronunciation
Definition
Meaning and Definition
The inverse of the secant function (sec⁻¹(x)), an inverse trigonometric function that returns the angle whose secant is a given number.
Also used as a unit of measurement in astronomy, equal to 1/3600 of a degree (an arc second). In mathematics and trigonometry, it's a standard notation for the inverse secant, a function whose domain is (-∞, -1] ∪ [1, ∞) and whose principal range is [0, π/2) ∪ (π/2, π].
Dialectal Variation
British vs American Usage
Differences
No significant lexical or conceptual differences. Both varieties treat it identically as a technical mathematical term. Potential minor spelling: 'arcsec' is standard; 'arc sec' (with space) is rare. Some older British texts might use 'inverse secant' more frequently than the abbreviation.
Connotations
Purely technical/neutral in both. No cultural or stylistic connotations attached.
Frequency
Equally rare in general usage. Used exclusively in technical domains (mathematics, physics, engineering). No regional frequency variation.
Grammar
How to Use “arcsec” in a Sentence
arcsec(x)y = arcsec xthe function arcsecVocabulary
Collocations
Usage
Meaning in Context
Business
Not used.
Academic
Used in university-level mathematics, physics, and engineering textbooks/courses, specifically in calculus, trigonometry, and related problem sets.
Everyday
Never used.
Technical
The only context of use. Found in mathematical proofs, physics calculations involving angles, and engineering specifications.
Vocabulary
Synonyms of “arcsec”
Strong
Neutral
Weak
Vocabulary
Antonyms of “arcsec”
Watch out
Common Mistakes When Using “arcsec”
- Using it with inputs between -1 and 1 (its domain excludes that interval).
- Confusing it with arccos, which is more common.
- Writing 'arc sec' as two words is less standard than 'arcsec'.
- Assuming its principal range is the same as arccos; it's different.
FAQ
Frequently Asked Questions
No, absolutely not. arcsec(x) = sec⁻¹(x) (the inverse function). 1/sec(x) is just cos(x). This is a common and serious misunderstanding.
Because the secant function is not one-to-one over its entire range. To create a proper inverse function, we restrict the domain of sec(x) to [0, π/2) ∪ (π/2, π], which then gives arcsec(x) its standard range.
Pronounced as 'ark-sek', with the stress on the first syllable. The 'c' is a hard 'k' sound.
Primarily in advanced mathematics (calculus, trigonometry), physics (especially optics and angular measurements), and certain branches of engineering. Its use as 'arcsecond' in astronomy is a separate, homographic abbreviation.
The inverse of the secant function (sec⁻¹(x)), an inverse trigonometric function that returns the angle whose secant is a given number.
Arcsec is usually formal technical/mathematical in register.
Learning
Memory Aids
Mnemonic
Think: 'ARC' gives you the ANGLE back. For SEC, you feed it a number (like 2) and ask, 'What ANGLE has a secant of 2?' The answer is arcsec(2). ARC = angle, SEC = secant.
Conceptual Metaphor
THE UNDO BUTTON: sec(x) takes an angle and gives a ratio. arcsec(x) is the 'undo' button—it takes a ratio and gives back the original angle.
Practice
Quiz
What is the principal value of arcsec(−2)?