arcsec
Definition
Noun:
- The inverse function of the secant: In trigonometry, arcsec is the inverse (or arc) function of the secant. For a given number x (where |x| ≥ 1), arcsec(x) returns the angle (typically measured in radians) whose secant is x.
Usage
- is used to find an angle when the value of its secant is known.
- It is commonly written as or .
- The output (the angle) is usually restricted to a specific range, most commonly [0, π] excluding π/2, to ensure it is a proper function.
Examples
Advanced Usage
- Domain and Range: The domain of is (-∞, -1] ∪ [1, ∞). Its principal value range is usually [0, π] except for π/2.
- Relationship with Arccosine: can be expressed in terms of the arccosine function: for x ≥ 1 or x ≤ -1.
- Derivative: In calculus, the derivative of is for |x| > 1.
Variants and Related Words
- Arcsecant: A full, alternative name for the function.
- sec⁻¹: An alternative notation for , where the "-1" denotes the inverse function, not the reciprocal.
- Inverse Secant: A descriptive term synonymous with .
Synonyms
- Inverse secant function: The formal descriptive synonym.
- Arc secant: A less common variant.
Notes
- It is crucial to distinguish or from , which is the reciprocal of the secant, i.e., the cosine: . The notation for the inverse function is a historical convention.
Noun
-
the inverse function of the secant; the angle that has a secant equal to a given number
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