Definition

1. Noun:
   • The inverse function of the tangent: arctan is a trigonometric function that, for a given number (representing a tangent value), returns the corresponding angle. Specifically, it yields the angle whose tangent is the given number. The result is typically an angle measured in radians.

Usage

• is used to find an angle when the value of the tangent of that angle is known.
• It is often denoted as or , where is a real number.
• The output (principal value) of the function is an angle in the range radians or .

Examples

• Noun/Function:
  • If tan(θ) = 1, then θ = arctan(1) = π/4 radians (or 45°).
  • To find the angle of elevation, calculate the arctan of the height divided by the distance.
  • The value arctan(0) is 0.

Advanced Usage

• Domain and Range: The function accepts any real number as input (its domain is all real numbers). Its output (range) is restricted to the interval to make it a proper function (single-valued).
• Relation to Complex Numbers: While primarily defined for real arguments, can be extended to handle complex numbers, though this is an advanced mathematical concept.
• In Calculus: The derivative of is , and its integral is .

Variants and Related Words

• arctangent (n): A full, synonymous name for .
• inverse tangent (n): A descriptive synonym for .
• tan⁻¹ (n): An alternative notation for , using exponent-like notation for the inverse function.
• atan (n): A common abbreviation used in computing and programming languages.

Synonyms

• inverse tangent: The descriptive term for the function.
• arctangent: The full name of the function.

Related Mathematical Functions

• arcsin: The inverse function of the sine.
• arccos: The inverse function of the cosine.
• arctan2 (or ): A common two-argument function in computing that computes the arctangent of , taking into account the quadrant of the point to return an angle in the full range .

Noun

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