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Arctangent (Arctan) Calculator

Compute arctan(x) — the inverse tangent — for any real number. Get the resulting angle in degrees, radians, or gradians instantly.

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Input Value (x)

Output Unit

Arctangent (angle)

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Arctangent (angle)—

The formula

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What Is the Arctangent Function?

The arctangent — written as arctan(x) or tan⁻¹(x) — is the inverse trigonometric function of the tangent. Given any real number x, the arcus tangent calculator returns the unique angle θ whose tangent equals x. The defining relationship is:

θ = arctan(x) if and only if tan(θ) = x

Because the tangent function repeats every 180°, it is not one-to-one across its full domain. Arctangent resolves this by restricting output to the principal value branch: the open interval (−90°, 90°), equivalently (−π/2, π/2) radians or (−100 grad, 100 grad). Every real-number input maps to exactly one output angle within that range.

Formula and Derivation

The formula at the core of every arctan calculation is:

θ = arctan(x)

For a right triangle, if the opposite side measures a units and the adjacent side measures b units, then:

θ = arctan(a / b)

This follows directly from the tangent ratio: tan(θ) = opposite / adjacent. Inverting that relationship yields arctangent. In calculus, the derivative plays a central role:

d/dx [arctan(x)] = 1 / (1 + x²)

Integrating 1 / (1 + x²) from 0 to 1 yields exactly π/4 ≈ 0.7854, a result that underpins Machin-type inverse tangent series for computing π to arbitrary precision.

Understanding the Variables

Input Value (x): Any real number on (−∞, +∞). Positive values of x produce first-quadrant angles between 0° and 90°; negative values produce fourth-quadrant angles between −90° and 0°. As x grows toward ±∞, arctan(x) approaches the horizontal asymptotes at ±90° but never reaches them.
Output Unit: Results can be expressed in degrees (°), radians (rad), or gradians (grad). Radians are standard in calculus, physics, and most programming libraries. Degrees appear in navigation and everyday geometry. Gradians divide a full right angle into exactly 100 equal parts, common in surveying.

Key Reference Values

arctan(0) = 0° = 0 rad
arctan(1) = 45° = π/4 rad ≈ 0.7854 rad
arctan(−1) = −45° = −π/4 rad
arctan(√3) = 60° = π/3 rad ≈ 1.0472 rad
arctan(1/√3) = 30° = π/6 rad ≈ 0.5236 rad
arctan(x) → 90° as x → +∞ (horizontal asymptote)

Real-World Applications

Navigation and Bearing

Pilots and surveyors use arctangent to compute bearing angles. An aircraft traveling 300 km east and 400 km north from its origin flies at a bearing of arctan(300 / 400) = arctan(0.75) ≈ 36.87° measured from north toward east.

Civil Engineering: Road Grade

A road that rises 5 m for every 100 m of horizontal run has a grade angle of arctan(5 / 100) = arctan(0.05) ≈ 2.86°. Engineers apply this angle when designing safe slopes and drainage gradients.

Physics: Force Vector Resolution

A 30 N horizontal force combined with a 40 N vertical force produces a resultant directed at arctan(40 / 30) ≈ 53.13° above horizontal. Resolving and reconstructing force vectors in this way is fundamental to structural and mechanical engineering.

Computer Graphics and Robotics

The two-argument variant atan2(y, x) extends arctangent to all four quadrants by using the individual signs of both coordinates, returning angles across the full (−180°, 180°] range. This function is indispensable in 2D/3D graphics engines and robotic motion planning.

Arctangent in Calculus and Mathematical Analysis

Beyond its geometric interpretation, arctangent appears frequently in advanced mathematics. The derivative 1 / (1 + x²) enables rapid convergence of infinite series approximations. Taylor series expansions and asymptotic methods rely on arctangent's well-behaved derivatives and predictable behavior at infinity. In complex analysis, the arctangent function extends into the complex plane, revealing deeper connections between inverse trigonometric functions and logarithmic expressions.

Using This Calculator

Enter any real number into the input field, select degrees, radians, or gradians, and the calculator applies θ = arctan(x) instantly. For equations of the form tan(θ) = k, the inverse trig equation method detailed by Paul's Online Math Notes shows that arctan gives the principal solution; additional solutions follow by adding integer multiples of 180° (or π rad) due to the tangent's periodicity. The theoretical basis for these calculations is covered in depth by the Xavier University of Louisiana inverse trig function reference.

Reference

Frequently asked questions

What is the difference between arctan(x) and tan⁻¹(x)?

Arctan(x) and tan⁻¹(x) are two notations for exactly the same function — the inverse tangent. The superscript −1 indicates an inverse function, not a reciprocal. The reciprocal of tan(x) is cotangent, cot(x) = 1/tan(x), which is an entirely different function. Both notations return the unique angle whose tangent equals x, always within the principal value range of (−90°, 90°) in degrees or (−π/2, π/2) in radians.

What is the domain and range of the arctangent function?

The domain of arctan(x) is all real numbers — every value on the interval (−∞, +∞) is a valid input. The range is the open interval (−90°, 90°) in degrees, (−π/2, π/2) in radians, or (−100, 100) in gradians. The function is strictly increasing throughout its domain and has horizontal asymptotes at ±90°, which it approaches but never reaches as x tends toward positive or negative infinity.

What is arctan(1) in degrees and radians?

Arctan(1) equals exactly 45° or π/4 radians, which is approximately 0.7854 rad. This value arises because tan(45°) = 1: in a 45-45-90 right triangle, the opposite and adjacent legs are equal in length, so their ratio equals 1. The value π/4 is also historically significant because integrating 1/(1+x²) from 0 to 1 gives π/4, connecting arctangent directly to the computation of π.

How do you convert an arctan result from radians to degrees?

Multiply the radian result by 180/π, which is approximately 57.2958. For example, arctan(1) = π/4 rad gives π/4 × (180/π) = 45°. To go the other direction, multiply degrees by π/180. For instance, 60° × (π/180) = π/3 rad ≈ 1.0472 rad. The arcus tangent calculator performs these conversions automatically when the output unit is changed, eliminating the need for manual multiplication.

Why does arctangent only return values between −90° and 90°?

The tangent function repeats with a period of 180°, meaning tan(θ) = tan(θ + 180°) for every angle θ. Because multiple angles share the same tangent value, the tangent function is not one-to-one and cannot have a true inverse over all angles. Restricting the output to the principal branch (−90°, 90°) — where tangent is strictly increasing and covers every real number exactly once — makes arctangent a well-defined, single-valued function with a unique output for every input.

What is the difference between arctan(x) and the atan2(y, x) function?

Arctan(x) accepts a single argument representing the ratio opposite/adjacent and returns a result confined to (−90°, 90°), so it cannot distinguish between angles in opposite quadrants that share the same ratio. The atan2(y, x) function takes the y and x coordinates separately, examines their individual signs, and returns a full-circle angle spanning (−180°, 180°]. This quadrant-aware behavior makes atan2 the standard choice in computer graphics, robotics, physics engines, and navigation systems where the correct quadrant must be identified.