Definition

1. Noun:
   • The inverse function of the cosine: A mathematical function that, for a given number (representing the cosine of an angle), returns the corresponding angle. Specifically, it yields the angle whose cosine is the given number. The output (the angle) is typically restricted to the principal value range of 0 to π radians (0 to 180 degrees).

Usage Examples

• Noun:
  • To find the angle in a right triangle when you know the adjacent side and hypotenuse, you use the inverse cosine.
  • The value of inverse cosine of 0.5 is π/3 radians, or 60 degrees.
  • Most calculators have a button labeled 'cos⁻¹' to calculate the inverse cosine.

Advanced Usage

• Mathematical Notation: The inverse cosine function is denoted as cos⁻¹(x), arccos(x), or arccosine(x). The notation "arccos" is often preferred to avoid confusion with the multiplicative inverse (1/cosine).
  • The solution is written as θ = arccos(0.707).
• Domain and Range: A critical concept is that the inverse cosine function is only defined for input values (x) in the domain [-1, 1]. Its output range is [0, π] in radians.
  • You cannot compute the inverse cosine of 2, as it lies outside the function's domain.

Variants and Related Words

• Arccosine (n): Another name for the inverse cosine function. It is the full, formal term.
  • "Arccosine" and "inverse cosine" are used interchangeably.
• cos⁻¹ (symbol): The common symbolic notation for the inverse cosine function.
• Inverse Trigonometric Function (n): The broader category of functions that includes inverse sine, inverse cosine, and inverse tangent.

Synonyms

• Arccos (n): Abbreviation for arccosine.
• Arccosine (n): A direct synonym.

Related Phrases and Concepts

• Principal Value: The inverse cosine returns the principal value of the angle, which is the unique angle within the range [0, π].
• Inverse Function: The inverse cosine is defined as the inverse of the cosine function, meaning if y = cos(θ), then θ = arccos(y) for θ in [0, π].

Noun

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