inverse sine

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Thân thiện
inverse sine

A student uses the inverse sine function to solve a geometry problem.

Definition
  1. Noun:
    • The inverse function of the sine: The inverse sine is a mathematical function that, for a given number (representing a sine value), returns the corresponding angle. It is the function that "undoes" the sine function.
    • The angle that has a sine equal to a given number: The inverse sine of a number is the angle (typically measured in radians or degrees) whose sine is that number.
Usage
  • The function is used to find an angle when the value of its sine is known.
  • It is commonly denoted as , , or .
  • The output (or range) of the function is restricted, usually to angles between -π/2 and π/2 radians (or -90° and 90°).
Examples
  • Noun:
    • To find the angle whose sine is 0.5, you calculate the inverse sine of 0.5.
    • The value of inverse sine(1/2) is π/6 radians or 30 degrees.
    • In solving the triangle, we used the inverse sine to determine the missing angle.
Advanced Usage
  • Principal Value: The function returns the , which is the angle within its restricted range.
    • For the input 0.707, the inverse sine returns the principal value of 45°, not 135°.
  • Notation and Ambiguity: The notation can be ambiguous, as it might be mistaken for (the cosecant). Context clarifies that it means the .
    • It's important to distinguish between sin⁻¹(x) for the inverse sine and (sin x)⁻¹ for the cosecant.
Variants and Related Words
  • arcsine (arcsin) (n): Another name for the function.
    • "Arcsine" is often used interchangeably with inverse sine.
  • inverse trigonometric function (n): A broader category of functions that includes , inverse cosine, and inverse tangent.
    • The inverse sine is one of the basic inverse trigonometric functions.
Synonyms
  • arcsine: The most direct synonym.
  • asin (abbreviation): A common abbreviation used in computing and mathematics.
Related Concepts (Not Phrasal Verbs or Idioms)
  • Domain and Range: The accepts inputs (domain) between -1 and 1 inclusive and outputs angles (range) between -π/2 and π/2.
    • The inverse sine is only defined for numbers in the interval [-1, 1].
  • Inverse Function: The is defined by the property: if , then for θ in the principal range.
inverse sine

A student uses the inverse sine function to solve a geometry problem.

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

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