Inverse Tangent Calculator (tan-1)

Enter a value to find the angle whose tangent equals that value, in degrees and radians.

Enter Value

45.0 deg

Result

arctan(value)
--
degrees
arctan(value)
--
radians
Input value--
Angle (degrees)--
Angle (radians)--
Quadrant--
sin of angle--
cos of angle--

Step-by-Step Solution

arctan(x) returns angle in (-90 deg, 90 deg)

Understanding Inverse Tangent (arctan)

The inverse tangent function, written as tan-1(x) or arctan(x), is the reverse of the tangent function. Given a value, it returns the angle whose tangent equals that value. The principal value range of arctan is (-90 degrees, 90 degrees) or (-pi/2, pi/2) radians.

Key Properties

Domain

The input can be any real number, from negative infinity to positive infinity.

Domain: (-inf, +inf)

Range

The output is always between -90 and 90 degrees (exclusive).

Range: (-90 deg, 90 deg)

Special Values

arctan(0) = 0, arctan(1) = 45 deg, arctan(-1) = -45 deg.

arctan(1) = 45 deg = pi/4

Unit Circle

On the unit circle, arctan(v) gives the angle where the y/x ratio equals v.

tan(theta) = y/x = v

Common arctan Values

  • arctan(0) = 0 degrees = 0 radians
  • arctan(1/sqrt(3)) = 30 degrees = pi/6 radians
  • arctan(1) = 45 degrees = pi/4 radians
  • arctan(sqrt(3)) = 60 degrees = pi/3 radians
  • arctan(+inf) = 90 degrees = pi/2 radians

Applications

The inverse tangent function is used extensively in engineering, physics, and computer graphics. It is essential for converting between rectangular and polar coordinates, calculating angles of inclination, and determining phase angles in electrical circuits.