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.

Noun

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