The Arctan Calculator computes the inverse tangent of any real value in degrees and radians. The only inverse trig function with no domain restriction — used in physics, engineering, and programming to recover angles from slopes, ratios, and component vectors.
45
°
0.785398
rad
45
°
0.785398
rad
While arcsin and arccos only accept inputs between −1 and +1, the arctangent accepts any real number — making it the most versatile inverse trigonometric function in practice. The calculator for arctan computes tan⁻¹(x) for any real input, returning the principal value angle in both degrees and radians, and includes the two-argument atan2(y, x) variant that correctly handles all four quadrants.
The arctangent function is the inverse of the tangent function, restricted to the principal value range:
arctan(x) = θ such that tan(θ) = x and −90° < θ < 90°
Unlike sine and cosine, tangent is defined for all real numbers (except odd multiples of 90°), so arctan has no domain restriction — any real input is valid. Key values:
As x → ±∞, arctan approaches ±90° asymptotically — the inverse tangent is bounded between −90° and +90° regardless of how large the input grows. The arcsin calculator and arccos calculator provide the complementary inverse trig functions.
The single-argument arctan(y/x) cannot distinguish between angles in different quadrants because (y/x) gives the same ratio for opposite quadrants: arctan(1/1) = arctan(−1/−1) = 45°, but the first point is in Q1 and the second in Q3. The two-argument function atan2(y, x) resolves this by using the signs of both y and x to determine the correct quadrant:
atan2 is used universally in programming (C, Python, MATLAB, JavaScript all implement it) and in vector angle calculations where the full 360° range must be correctly determined. Use this online calculator for both the single and two-argument variants.
Arctan is the go-to function for recovering angles from slopes, ratios, and component vectors:
The reference angle calculator and trigonometry calculators provide the complete inverse trig toolkit.
The arctangent function has a beautiful power series representation (Gregory-Leibniz series): arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + ... for |x| ≤ 1. Setting x = 1 gives the famous Leibniz formula for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + ... — one of the simplest known series for π, though it converges extremely slowly. Modern fast arctan series for π use the Machin formula: π/4 = 4·arctan(1/5) − arctan(1/239), which converges much faster because arctan(x) is well approximated by x for small x.
Enter any real number. The calculator applies the inverse tangent function to determine the angle whose tangent equals your input value. Results are shown in both degrees and radians.
The result is the angle in the range $$(-90°, 90°)$$ whose tangent equals your input. For large positive inputs, the angle approaches 90°; for large negative inputs, it approaches -90°. An input of 0 always gives 0°, and an input of 1 gives exactly 45°.
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Since tan(45°) = 1, the arctangent of 1 is exactly 45° or π/4 radians.
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Since tan(−60°) = −√3 ≈ −1.732, the result is −60° or −π/3 radians.
$$\arctan(x)$$ takes a single value and returns an angle in $$(-90°, 90°)$$. The function $$\text{atan2}(y, x)$$ takes two arguments (the y and x coordinates) and returns the full-circle angle in $$(-180°, 180°]$$, correctly handling all four quadrants. Use atan2 when you need to determine direction from coordinates.
The tangent function can produce any real number as output (its range is all of $$\mathbb{R}$$). Since arctangent is the inverse of tangent, its domain is all real numbers. This contrasts with arcsine and arccosine, whose domains are limited to $$[-1, 1]$$.
$$\arctan(0) = 0°$$ or $$0$$ radians, because $$\tan(0°) = 0$$.
If a surface rises by $$r$$ units over a horizontal distance of $$d$$ units, the slope angle is $$\theta = \arctan\left(\frac{r}{d}\right)$$. For example, a 1:1 slope (rise equals run) gives $$\arctan(1) = 45°$$.
As $$x \to +\infty$$, $$\arctan(x)$$ approaches $$90°$$ ($$\frac{\pi}{2}$$) but never reaches it. Similarly, as $$x \to -\infty$$, it approaches $$-90°$$. These horizontal asymptotes give the arctangent curve its characteristic S-shape.
$$\int \arctan(x)\,dx = x\,\arctan(x) - \frac{1}{2}\ln(1 + x^2) + C$$. This is derived using integration by parts with $$u = \arctan(x)$$ and $$dv = dx$$.
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