arcsecant

Thân thiện

The student calculates the arcsecant of a value on her calculator.

Definition

1. Noun:
   • The inverse function of the secant: In trigonometry, the arcsecant is the inverse function of the secant. For a given number x where |x| ≥ 1, the arcsecant of x is the angle (typically measured in radians) whose secant is x. It returns an angle in the range [0, π] excluding π/2.

Usage

• The arcsecant function is used to find an angle when the value of the secant of that angle is known.
• It is denoted as arcsec(x) or sec⁻¹(x).
• The input value () must satisfy || ≥ 1 because the secant of an angle is never between -1 and 1.

Examples

• Noun:
  • To solve the equation sec(θ) = 2, we calculate θ = arcsec(2).
  • The value of arcsecant(-√2) is 3π/4 radians.
  • The derivative of the arcsecant function is an important result in calculus.

Advanced Usage

• Principal Value: The principal value of is conventionally defined to be in the interval [0, π], except for π/2, which is not in the range of the arcsecant.
• Relation to Arccosine: The arcsecant can be expressed in terms of the arccosine: for || ≥ 1.

Variants and Related Words

• arcsec (abbr.): The standard abbreviation for arcsecant.
  • The angle is arcsec(4).
• Inverse Secant: A synonymous phrase for arcsecant.
• Secant Inverse: An alternative, less common phrasing.

Synonyms

• Inverse secant function: The full descriptive name.
• sec⁻¹: The standard mathematical notation.

Notes

• The arcsecant is a transcendental function.
• It is one of the six principal inverse trigonometric functions, along with arcsine, arccosine, arctangent, arccosecant, and arccotangent.
• Handling the domain and range correctly is crucial, as the secant function is not one-to-one over its entire domain; its domain is restricted to [0, π] (excluding π/2) to define the inverse.

Học thuật

Noun

1. the inverse function of the secant; the angle that has a secant equal to a given number

Từ đồng nghĩa

• arc secant
• arcsec
• inverse secant