arc cosecant
Definition
- Noun:
- The inverse function of the cosecant: The
arc cosecantof a number is the angle (usually measured in radians or degrees) whose cosecant equals that given number. It is the principal value of the inverse cosecant function.
Usage
- The is used in trigonometry and calculus to find an angle when the value of the cosecant is known.
- It is typically denoted as arccsc(x), csc⁻¹(x), or arc cosec(x).
- The domain of the function is (-∞, -1] ∪ [1, ∞), and its principal value range is usually [-π/2, 0) ∪ (0, π/2] (in radians).
Examples
Advanced Usage
- Mathematical Context: The is one of the six principal inverse trigonometric functions. Its derivative is d/dx[arccsc(x)] = -1 / (|x|√(x² - 1)) for |x| > 1.
- Principal Value: It is important to specify the range (or branch) when discussing the , as it is a multivalued function. The principal value is the most commonly used output.
Variants and Related Words
- arccsc: The standard abbreviation for .
- inverse cosecant: A synonymous term for .
- arcsecant (arcsec): The inverse of the secant function, related conceptually.
- arcsine (arcsin): The inverse of the sine function, another inverse trigonometric function.
Synonyms
- Inverse cosecant: The direct synonym.
- arccsc: The standard mathematical abbreviation.
Related Phrases and Concepts
- Inverse trigonometric function: The broader category to which belongs.
- Cosecant (csc): The trigonometric function whose inverse is the .
Noun
-
the angle that has a cosecant equal to a given number
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