arctan15

Arctan15, formally denoted as arctan(15), is the inverse trigonometric function that returns the angle whose tangent is 15. In mathematical notation it is expressed as θ = arctan(15), where θ is measured in radians. The value of arctan(15) is approximately 1.50459 radians, which is about 86.19 degrees. This angle lies in the first quadrant, as the tangent function is positive for angles between 0 and π/2.

Because the tangent function increases monotonically from 0 to π/2, the inverse function arctan is single‑valued on the real line. This makes arctan(15) a well‑defined real constant. In many scientific and engineering contexts the arctangent of large arguments such as 15 is used in geometry, signal processing, and control theory when converting between slope ratios and angular measurements.

The calculation of arctan(15) can be performed using power‑series expansions, continued fractions, or polynomial approximations such as the Chebyshev or Remez algorithms. Most modern computing libraries provide an arctan function that directly evaluates this value to double‑precision accuracy. When high precision is required, multiprecision arithmetic libraries can compute arctan(15) to thousands of decimal places, with the result used in symbolic mathematics or in the verification of analytic expressions.

Although arctan(15) is not a special angle like π/4 or π/6, it frequently appears in problems that involve the ratio of a vertical side to a horizontal side equal to 15, for example when determining the elevation angle of a target that is 15 units up for each unit of horizontal distance. In such applications the arctangent function provides the necessary conversion from a ratio to an angular representation.