Skip to main content
math

Arctan Calculator (Inverse Tangent)

Calculate the inverse tangent (arctan) of a value. Returns the angle whose tangent equals the input.

By Christopher FloiedPublished Updated

The tangent value (any real number)

Results

Angle (degrees)

45°

Angle (radians)

0.785398 rad

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Arctan Calculator (Inverse Tangent). Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.

2

Review your inputs

Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Arctan Calculator (Inverse Tangent) computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Arctan Calculator (Inverse Tangent) when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

Related Calculators

Arccos Calculator (Inverse Cosine)

Calculate the inverse cosine (arccos) of a value. Returns the angle whose cosine equals the input.

Arcsin Calculator (Inverse Sine)

Calculate the inverse sine (arcsin) of a value. Returns the angle whose sine equals the input.

Cosine Calculator

Calculate the cosine of any angle in degrees or radians. Returns values between -1 and 1.

Cotangent Calculator

Calculate the cotangent of any angle. Cotangent is the reciprocal of tangent: cot(θ) = cos(θ)/sin(θ).

Sine Calculator

Calculate the sine of any angle in degrees or radians. Returns values between -1 and 1.

Tangent Calculator

Calculate the tangent of any angle in degrees. The tangent is the ratio sin(θ)/cos(θ).

About Arctan Calculator (Inverse Tangent)

The arctangent function (also written tan⁻¹ or arctan) is the inverse of the tangent function. Unlike arcsine and arccosine, arctan accepts any real number as input and returns an angle in the range (−90°, 90°) or (−π/2, π/2) radians. This function is used extensively in navigation to calculate bearing angles, in computer science for the atan2 function that determines direction, in control theory for phase margin calculations, and in calculus for evaluating integrals involving 1/(1 + x²). The arctangent is also the basis for many formulas for computing π, including Machin's formula: π/4 = 4 arctan(1/5) − arctan(1/239). This calculator provides the angle for any tangent value in both degrees and radians.

The Math Behind It

The tangent function is restricted to (−π/2, π/2) for its inverse, where it is strictly increasing and covers all real numbers. The derivative of arctan is d/dx arctan(x) = 1/(1 + x²), one of the cleanest derivatives in calculus. This leads to the integral ∫ dx/(1 + x²) = arctan(x) + C. The Taylor series is arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + ..., which converges for |x| ≤ 1. Setting x = 1 gives the Leibniz formula: π/4 = 1 − 1/3 + 1/5 − 1/7 + ..., though this converges very slowly. The two-argument function atan2(y, x) extends arctan to cover all four quadrants, returning angles in (−180°, 180°]. It is defined as the angle of the vector (x, y) and handles the sign ambiguity that single-argument arctan cannot resolve. The Gudermannian function gd(x) = 2 arctan(tanh(x/2)) connects hyperbolic and circular functions without complex numbers and appears in the Mercator map projection.

Formula Reference

Inverse Tangent

θ = arctan(x) = tan⁻¹(x)

Variables: x ∈ (−∞, ∞), θ ∈ (−90°, 90°)

Worked Examples

Example 1: Slope to angle

A road has a slope (rise/run) of 0.15. Find the angle of inclination.

Step 1:tan(θ) = 0.15
Step 2:θ = arctan(0.15)

θ ≈ 8.53°

Common Mistakes & Tips

  • !Forgetting that arctan returns values in (−90°, 90°) — use atan2 for full 360° range.
  • !Confusing tan⁻¹(x) with 1/tan(x) (which is cot(x)).
  • !Not considering that two different angles can have the same tangent value (differing by 180°).

Related Concepts

Arcsin Calculator

Inverse sine function.

Tangent Calculator

The forward tangent function.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Arcsin Calculator (Inverse Sine)

Tangent Calculator

Frequently Asked Questions

What is the difference between arctan and atan2?

arctan(x) takes one argument and returns angles in (−90°, 90°). atan2(y, x) takes two arguments and returns angles in (−180°, 180°], correctly handling all four quadrants.

Can arctan be used to compute π?

Yes. Since arctan(1) = π/4, we get π = 4 arctan(1). More efficient formulas like Machin's formula use combinations of arctan values.

Add the live Arctan Calculator (Inverse Tangent) to your site. Free to embed, with a link back to MegaCalc.