Definition

1. Noun:
   • The inverse function of the cosine: arccos is a trigonometric function that returns the angle whose cosine is a given number. It is the principal value of the inverse cosine function.

Usage

• is used in mathematics, physics, and engineering to find an angle when the cosine of that angle is known.
• The input value (the cosine) must be between -1 and 1, inclusive. The output (the angle) is typically given in radians, ranging from 0 to π (0 to 180 degrees).
• It is commonly written as or .

Examples

• Noun:
  • If cos(θ) = 0.5, then θ = arccos(0.5) = π/3 radians (or 60 degrees).
  • To find the angle in the triangle, we calculated the arccos of the adjacent side divided by the hypotenuse.

Advanced Usage

• Principal Value: The function is defined to return a single, specific value (the principal value) for each input to make it a proper function. For a real number where -1 ≤ x ≤ 1, is the unique angle in the interval [0, π] such that cos(y) = x.
• Relation to Arc Cosine: The name "arccos" is an abbreviation of "arcus cosine," meaning "the arc whose cosine is" the given number.

Variants and Related Words

• arccosine (n): A full, alternative name for . It means the same thing.
  • The arccosine of 0 is π/2.
• inverse cosine (n): A descriptive term synonymous with .
  • The inverse cosine function is denoted as cos⁻¹.
• acos (n): A common abbreviation used in programming languages and calculators.
  • In the code, use the acos() function to compute the angle.

Synonyms

• inverse cosine: The function that undoes the cosine operation.
• cos⁻¹: The standard mathematical notation for the inverse cosine function.

Related Phrases/Concepts

• Domain and Range: A key concept when using . Its domain is [-1, 1] and its range is [0, π] (in radians).
• Trigonometric Identity: is related to other inverse functions. For example, arccos(x) = π/2 - arcsin(x) for x in [-1, 1].

Noun

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