Arctan Calculator (Inverse Tangent)

Arctangent answers 'what angle has this tangent?', returning a value between −90° and 90°.

How the Math Works

The arctangent function, denoted as arctan(x) or tan⁻¹(x), is the inverse of the tangent function. It calculates the angle θ in a right triangle whose tangent (opposite/adjacent) equals the input value x. Mathematically, if tan(θ) = x, then θ = arctan(x). The calculator accepts any real number input and outputs the angle in radians (default) or degrees, typically within the range of -π/2 to π/2 (or -90° to 90°), representing the principal value of the angle. This function is fundamental in trigonometry for solving angle measurements when only ratio information is available.

Practical Applications

Arctan calculations are essential in engineering and physics for determining angles in structural analysis, such as calculating the slope of a beam or the angle of a ramp. In navigation and robotics, arctan helps compute bearings and heading directions using coordinate differences (e.g., Δy/Δx). Computer graphics rely on arctan for rotation transformations, where objects are rotated by specific angles derived from coordinate vectors. Surveyors use it to compute elevation angles when measuring heights or distances, and electrical engineers apply it in AC circuit analysis to determine phase differences between voltage and current waveforms.

Day-to-Day Use

In everyday scenarios, arctan aids in practical problem-solving: setting up ladders at safe angles (typically 75° from the ground), adjusting solar panel orientations for optimal sunlight exposure, or calculating the correct angle to aim a sports shot (e.g., pool or golf). DIY enthusiasts use it to ensure precise cuts in carpentry or construction projects. Even in video games, arctan helps simulate realistic camera movements and character rotations based on player input. It also appears in finance when analyzing logarithmic returns or volatility indices, making it a versatile tool beyond pure mathematics.

Worked example

arctan(1) = 45°.

FAQ