Arctan Calculator
Calculate the inverse tangent ($\arctan(x)$ or $\tan^{-1}(x)$) for any real number with exact $\pi$ radians, decimal degrees, slope grade percentage, and 4-quadrant $\text{atan2}(y, x)$ coordinate analysis.
Unit Circle & Tangent Line Visualizer
Range: (−π/2, π/2)Step-by-Step Arctan Evaluation & Range Analysis Range: −π/2 < θ < π/2 (−90° < θ < 90°)
How to Calculate Arctan (Inverse Tangent)
To find the arctan of a number x, determine the principal angle θ between -π/2 and +π/2 (-90° to +90°) whose tangent equals x: θ = arctan(x) ⟺ tan(θ) = x. Because tangent is opposite over adjacent (or rise over run), arctan(x) geometrically represents the angle of a line with slope m = x. For standard values, arctan(1) = 45° (π/4 rad), arctan(√3) = 60° (π/3 rad), arctan(1/√3) = 30° (π/6 rad), and arctan(0) = 0°.
Anatomy of the Arctan Function & Slope Interpretation
The inverse tangent (written as $\arctan(x)$ or $\tan^{-1}(x)$) is the mathematical operation that reverses the tangent function. While $\tan(\theta)$ takes an angle and produces a ratio of side lengths ($\frac{\text{opposite}}{\text{adjacent}}$), $\arctan(x)$ takes a ratio and returns the original angle $\theta$.
In Cartesian coordinate geometry, the slope $m$ of a line is defined as $\frac{\Delta y}{\Delta x} = \tan(\theta)$. Therefore, the angle of inclination of any line with slope $m$ is given directly by $\theta = \arctan(m)$.
Accepts any real number without restriction.
Restricted to Quadrants I and IV (−90° to 90°).
Symmetric about the origin (rotational symmetry).
Domain (-∞, ∞) & Principal Range (-π/2, π/2)
Because the periodic tangent function $\tan(\theta)$ repeats every $\pi$ radians ($180^\circ$), it is not one-to-one across its natural domain. To define a true single-valued inverse mathematical function, mathematicians restrict the domain of $\tan(\theta)$ to the principal open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
θ = arctan(x) ⇔ tan(θ) = x where −π/2 < θ < π/2
As $x \to +\infty$, $\arctan(x) \to \frac{\pi}{2}^-$ ($+90^\circ$).
As $x \to -\infty$, $\arctan(x) \to -\frac{\pi}{2}^+$ ($-90^\circ$).
Exact Values Reference Table (Radicals & Degrees)
| Input Ratio x | Exact Radians θ | Degrees θ | Decimal Radians | Quadrant |
|---|---|---|---|---|
| −√3 | −π/3 | −60° | −1.0472 | Quadrant IV |
| −1 | −π/4 | −45° | −0.7854 | Quadrant IV |
| −√3/3 | −π/6 | −30° | −0.5236 | Quadrant IV |
| 0 | 0 | 0° | 0.0000 | Origin / X-Axis |
| √3/3 | π/6 | 30° | 0.5236 | Quadrant I |
| 1 | π/4 | 45° | 0.7854 | Quadrant I |
| √3 | π/3 | 60° | 1.0472 | Quadrant I |
arctan(y/x) vs 4-Quadrant atan2(y, x) in Programming
In computer programming (JavaScript, Python, C++, Java), calculating polar angles using standard $\arctan(y/x)$ introduces severe ambiguity:
Point (3, 3) in Quad I → y/x = 1 → arctan(1) = 45°.
Point (-3, -3) in Quad III → y/x = 1 → arctan(1) = 45°! (Incorrect angle by 180°).
atan2(3, 3) = +45° (+π/4 rad, Quadrant I).
atan2(-3, -3) = -135° (-3π/4 rad, Quadrant III).
Calculus of Arctan: Derivatives & Taylor Series
With chain rule: d/dx [arctan(u)] = (du/dx) / (1 + u²).
Derived using integration by parts with u = arctan(x), dv = dx.
Step-by-Step Worked Solutions
A ramp rises 5 meters over a horizontal run of 12 meters. What is the angle of elevation θ?
1. Slope ratio x = opposite / adjacent = 5 / 12 ≈ 0.41667.
2. Apply arctan: θ = arctan(5/12) ≈ 0.3948 radians.
3. Convert to degrees: θ = 0.3948 × (180 / π) ≈ 22.620°.
Common Pitfalls: tan⁻¹(x) vs Reciprocal cot(x)
In mathematical notation, $\tan^{-1}(x)$ denotes the inverse function ($\arctan x$), NOT $\frac{1}{\tan x} = \cot(x)$. For example, $\tan^{-1}(1) = 45^\circ$, whereas $(\tan 45^\circ)^{-1} = 1/1 = 1$.
Always verify whether your handheld calculator is set to Radian or Degree mode before writing exam answers.
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.