Tan Inverse
Calculate inverse tangent (arctan/tan⁻¹) of any real number. Get the angle in degrees and radians with an interactive arctan graph and key value table.
About This Calculator
The inverse tangent function, denoted tan⁻¹(x) or arctan(x), is the inverse of the tangent function. It returns the angle whose tangent equals the input value x. Unlike arcsin and arccos which are restricted to [-1, 1], arctan accepts any real number from -∞ to +∞. The principal value lies between -90 deg and 90 deg (-pi/2 to pi/2 radians), approaching these horizontal asymptotes as x approaches +/-∞ but never reaching them.
The mathematical relationship is: theta = arctan(x) where tan(theta) = x. This calculator provides both degree and radian outputs simultaneously, making it essential for students studying trigonometry, calculus, and advanced mathematics. Key values include: arctan(0) = 0 deg, arctan(1) = 45 deg (pi/4 rad), arctan(sqrt3) = 60 deg (pi/3 rad), and arctan(-1) = -45 deg (-pi/4 rad). The function is odd, meaning arctan(-x) = -arctan(x).
The arctan function is related to other trigonometric functions through fundamental identities. The derivative of arctan(x) is 1/(1 + x^2), and its integral is x·arctan(x) - 1/2·ln(1 + x^2) + C. The two-argument variant atan2(y, x) extends arctan to return the angle from the positive x-axis for any point (x, y), correctly handling all four quadrants. This is widely used in computer programming for coordinate transformations.
Inverse tangent is fundamental in computer science (the atan2 function used in game development, computer graphics for camera rotation, and robotics for inverse kinematics), civil engineering (calculating road gradients, ramp slopes, and roof pitches), surveying (angle of elevation and depression measurements), geography (computing bearings from coordinate differences and great-circle navigation), and machine learning (as activation functions in neural networks and for feature engineering with angular transformations).
Students encounter arctan in trigonometry when solving for angles in right triangles, in calculus when integrating rational functions, and in physics when resolving vectors and calculating projectile trajectories. Unlike the tangent function which repeats every 180 deg, arctan is a single-valued function with a well-defined principal value, making it unambiguous for practical computations.
Frequently Asked Questions
What is tan⁻¹ or inverse tangent?
Tan inverse (arctan) is the inverse function of tangent. If tan(theta) = x, then arctan(x) = theta. It returns the angle whose tangent equals the given number. The output range is -90 deg to 90 deg (-pi/2 to pi/2 radians), unlike arcsin and arccos.
What values can I input for tan inverse?
Unlike arcsin and arccos, arctan accepts any real number input from -∞ to +∞. The output always falls between -90 deg and 90 deg (-pi/2 to pi/2 radians). This makes it useful for converting slopes and ratios into angles.
How does arctan differ from arcsin and arccos?
Arctan accepts any real number input, while arcsin and arccos are restricted to [-1, 1]. Arctan outputs between -90 deg and 90 deg, arcsin also outputs -90 deg to 90 deg, but arccos outputs 0 deg to 180 deg. Unlike the others, arctan has horizontal asymptotes at +/-90 deg.
Where is inverse tangent used in real life?
Inverse tangent is essential for converting slopes to angles in civil engineering (road grades, roof pitches), in computer graphics for camera rotation (atan2 function), in navigation (bearing from coordinates), in robotics (inverse kinematics), and in machine learning (activation functions).
What is the atan2 function?
Atan2(y, x) is a two-argument variant of arctan that returns the angle from the positive x-axis to the point (x, y). Unlike arctan(y/x), atan2 correctly handles all four quadrants and avoids division by zero, returning angles from -pi to pi.
Is this tan inverse calculator free?
Yes, all calculators on Calculy are completely free to use. No registration or download required.
What is the derivative of arctan?
The derivative of arctan(x) is 1/(1 + x^2). This derivative exists at every real number, making the arctan function differentiable everywhere. The derivative is derived from the inverse function theorem and is fundamental in integral calculus, where it forms the basis for integrating rational functions of the form 1/(a^2 + x^2).
What is the integral of arctan?
The integral of arctan(x) is x·arctan(x) - 1/2·ln(1 + x^2) + C. This formula is derived using integration by parts, where u = arctan(x) and dv = dx. It appears frequently in calculus courses and is used in probability theory for computing certain integrals involving the Cauchy distribution.