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Inverse Tangent Calculator — Definition, Formula & Examples

An inverse tangent calculator takes a number (the ratio of opposite to adjacent sides) and returns the angle whose tangent equals that number. The output always falls between −90°-90° and 90°90° (or −π2-\frac{\pi}{2} and π2\frac{\pi}{2} radians).

Given a real number xx, the inverse tangent function arctan⁡(x)\arctan(x) returns the unique angle θ∈(−π2, π2)\theta \in \left(-\frac{\pi}{2},\, \frac{\pi}{2}\right) such that tan⁡(θ)=x\tan(\theta) = x. An inverse tangent calculator evaluates this function numerically for any real input.

Key Formula

θ=arctan⁡(x)where tan⁡(θ)=x\theta = \arctan(x) \quad \text{where } \tan(\theta) = x
Where:
  • xx = Any real number (the tangent ratio you are finding the angle for)
  • θ\theta = The resulting angle, always in $\left(-\frac{\pi}{2},\, \frac{\pi}{2}\right)$

How It Works

Enter any real number into the calculator — there are no domain restrictions for arctan, unlike arcsin or arccos. The calculator returns an angle in the range (−90°,90°)(-90°, 90°) if set to degrees, or (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) if set to radians. On a scientific calculator, press the "tan⁻¹" or "atan" button after typing the value. To convert a radian result to degrees, multiply by 180π\frac{180}{\pi}.

Worked Example

Problem: Find the angle whose tangent is 1. Express the answer in both degrees and radians.
Set up: You need θ\theta such that tan⁡(θ)=1\tan(\theta) = 1.
θ=arctan⁡(1)\theta = \arctan(1)
Evaluate: From the unit circle, tan⁡(45°)=1\tan(45°) = 1, and 45°45° lies within the output range.
θ=45°=π4 rad\theta = 45° = \frac{\pi}{4} \text{ rad}
Answer: arctan⁡(1)=45°\arctan(1) = 45° or π4\frac{\pi}{4} radians.

Why It Matters

You use arctan whenever you know a slope or a rise-over-run ratio and need the corresponding angle — common in physics (projectile launch angles), engineering (ramp design), and navigation (bearing calculations). It also appears in calculus when integrating 11+x2\frac{1}{1+x^2}.

Common Mistakes

Mistake: Forgetting to check whether your calculator is in degree mode or radian mode before reading the output.
Correction: Always verify the mode first. If you need degrees but get 0.78540.7854, that is π4\frac{\pi}{4} radians — multiply by 180π\frac{180}{\pi} to convert to 45°45°.

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