atan20

atan20 refers to the arctangent of the real number 20, i.e., the value of the inverse tangent function at 20. In standard notation it is arctan(20) or tan^(-1)(20). Some calculators or programming contexts may display the result as atan(20), and in informal shorthand it can be written as atan20. The arctangent function has domain all real numbers and range (-pi/2, pi/2); for positive inputs it yields a positive angle in the first quadrant.

Numerical value and form: arctan(20) is approximately 1.52084 radians, which is about 87.14 degrees. A useful identity is arctan(20) = pi/2 − arctan(1/20). This follows from the complementary relationship of arctan for positive arguments and often helps with alternative calculations.

Properties: The derivative of arctan x with respect to x is 1/(1+x^2). At x = 20, the derivative is 1/401 ≈ 0.00249, reflecting a relatively gentle slope near large positive x. As x → ∞, arctan x approaches pi/2, and as x → −∞ it approaches −pi/2.

Notation and usage: While “atan20” can be encountered as shorthand, it is not a distinct mathematical function; it denotes evaluating the standard arctan function at the argument 20. In programming, arctan is typically provided as atan or atan2, the latter handling two-argument cases for coordinated angle determination.

Applications: arctan(20) appears in contexts involving slope-to-angle conversions, orientation calculations, and various trigonometric computations in science, engineering, and computer graphics.

See also: arctan, arctan2, tangent.