Understanding Inverse Tangent (Arctan)
The inverse tangent function, written as arctan(x) or tan-1(x), returns the angle whose tangent is x. Unlike arcsin and arccos, arctan accepts any real number as input, since the tangent function can produce any real value. The output is restricted to (-90°, 90°) or (-π/2, π/2) radians.
Common Arctan Values
arctan(0) = 0°
The tangent of 0 is 0. This is the center of the arctan function.
arctan(√3/3) = 30°
From the 30-60-90 triangle: tan(30°) = 1/√3 = √3/3.
arctan(1) = 45°
The most well-known value. tan(45°) = 1 from the isosceles right triangle.
arctan(√3) = 60°
From the 30-60-90 triangle: tan(60°) = √3.
arctan vs atan2
The standard arctan function returns angles in (-90°, 90°), covering only quadrants I and IV. The atan2(y, x) function is a two-argument variant that considers the signs of both y and x to determine the correct quadrant, returning angles in (-180°, 180°]. This makes atan2 essential for applications where the full angle range matters, such as navigation, robotics, and computer graphics.
Key Properties
- arctan(x) is an increasing function defined for all real numbers.
- As x approaches infinity, arctan(x) approaches 90° (π/2).
- As x approaches negative infinity, arctan(x) approaches -90° (-π/2).
- arctan(-x) = -arctan(x) (odd function).
- arctan(1/x) = π/2 - arctan(x) for x > 0.
- The derivative: d/dx arctan(x) = 1/(1 + x²).
- The integral of 1/(1 + x²) is arctan(x) + C.
Applications
- Computer graphics: Converting between Cartesian and polar coordinates.
- Robotics: Calculating joint angles and heading directions.
- Navigation: Bearing calculations between two GPS coordinates.
- Signal processing: Computing phase angles of complex signals.
- Physics: Finding angles in force decomposition problems.