Arctan Calculator

Use this free arctan calculator by entering any real number as the tangent value. The calculator instantly returns the angle in both degrees and radians with step-by-step explanation.

The arctan formula is: arctan(x) = θ where tan(θ) = x and -π/2 < θ < π/2 (-90° < θ < 90°). It is the inverse function of tangent.

Arctan (inverse tangent) is the function that takes a tangent value and returns the angle whose tangent is that value. It is written as arctan(x) or tan⁻¹(x) and produces angles between -90° and 90° (exclusive).

The domain of arctan is all real numbers (-∞, +∞). The range is (-π/2, π/2) in radians or (-90°, 90°) in degrees. The endpoints are excluded because tan(±90°) is undefined.

Yes, this free online arctan calculator is 100% free with no signup required. Calculate inverse tangent values instantly.

arctan(1) = 45° or π/4 radians ≈ 0.7854. This is because tan(45°) = 1.

arctan(0) = 0° or 0 radians. Since tan(0) = 0, the inverse tangent of 0 is 0.

arctan(√3) = 60° or π/3 radians ≈ 1.0472. Since tan(60°) = √3.

arctan(1/√3) = 30° or π/6 radians ≈ 0.5236. Since tan(30°) = 1/√3.

arctan(-1) = -45° or -π/4 radians ≈ -0.7854. The arctan function is odd, so arctan(-x) = -arctan(x).

arctan(∞) approaches π/2 radians (90°) but never reaches it. As x → +∞, arctan(x) → π/2 from below.

arctan(-∞) approaches -π/2 radians (-90°) but never reaches it. As x → -∞, arctan(x) → -π/2 from above.

Multiply the radian value by 180/π. For example, arctan(1) = π/4 ≈ 0.7854 radians × (180/π) = 45°.

Multiply the degree value by π/180. For example, 45° × (π/180) = π/4 ≈ 0.7854 radians.

d/dx[arctan(x)] = 1/(1+x²). This is a fundamental calculus result used in integration and differentiation.

∫1/(1+x²)dx = arctan(x) + C. This is one of the standard integral forms in calculus.

arctan(tan(θ)) = θ only when θ is in the principal range (-π/2, π/2). Outside this range, the result wraps back into it.

tan(arctan(x)) = x for all real numbers x. Since arctan and tan are inverse functions, composing them cancels out.

arctan(x) + arccot(x) = π/2 for all real x. They are complementary angles.

arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ... for |x| ≤ 1. This converges slowly for values near ±1.

For large |x|, arctan(x) ≈ π/2 - 1/x (for x > 0). The calculator handles floating-point precision well for all real inputs.

Arctan is used in navigation (bearing calculations), robotics (angle from coordinates), physics (phase angles), and engineering (signal processing).

Yes. If you know the opposite and adjacent sides, arctan(opposite/adjacent) gives the angle. For example, opposite=3, adjacent=4: arctan(3/4) ≈ 36.87°.

arctan(2) ≈ 63.435° or approximately 1.1071 radians.

arctan(0.5) ≈ 26.565° or approximately 0.4636 radians.

For small x, arctan(x) ≈ x. The calculator maintains full floating-point precision even for tiny values.

Yes, arctan and tan⁻¹ are the same function. Both denote the inverse tangent function. The notation tan⁻¹(x) is more common in textbooks.

The range of arctan in degrees is (-90°, 90°), exclusive of the endpoints. It never equals exactly ±90°.

The range of arctan in radians is (-π/2, π/2), exclusive of the endpoints. This is approximately (-1.5708, 1.5708).

No. The principal value of arctan is always between -90° and 90°. For angles outside this range, use atan2 or adjust manually.

atan2(y, x) considers the signs of both y and x to determine the correct quadrant, returning values from -π to π. arctan(y/x) loses sign information.

Slope angle = arctan(rise/run). For a 45° slope, rise = run, so arctan(1) = 45°. For a 100% grade, arctan(1) = 45°.

In complex numbers, arctan helps find the argument (angle) of a complex number z = a + bi: arg(z) = arctan(b/a) with quadrant adjustment.

arctan(-x) = -arctan(x). The function is odd symmetric about the origin.

d/dx[arctan(u)] = u'/(1+u²). For example, d/dx[arctan(3x)] = 3/(1+9x²).

arctan(0.577) ≈ 30° since 0.577 ≈ 1/√3.

arctan(1.732) ≈ 60° since 1.732 ≈ √3.

Use math.atan(x) for single values or numpy.arctan(x) for arrays. The function returns radians by default.

Use the ATAN function: =ATAN(number). The result is in radians. Multiply by 180/PI() to convert to degrees.

arctan takes a ratio (opposite/adjacent) and returns an angle. arccos takes a cosine value (-1 to 1) and returns an angle (0° to 180°).

arctan takes any real number and returns angle in (-90°, 90°). arcsin takes values in [-1, 1] and returns angle in [-90°, 90°].

Mathematically yes: arctan(z) = i/2 · ln((1-iz)/(1+iz)). This calculator handles real inputs only.

arctan(10) ≈ 84.289° or approximately 1.4711 radians.

arctan(0.1) ≈ 5.711° or approximately 0.0997 radians.

The calculator uses JavaScript's Math.atan() function, which implements IEEE 754 double-precision floating-point arithmetic for high accuracy.

arctan(x) = Σ(-1)ⁿ · x^(2n+1)/(2n+1) for n=0 to ∞. This converges for |x| ≤ 1.

For x > 0: arctan(x) + arctan(1/x) = π/2. For x < 0: arctan(x) + arctan(1/x) = -π/2.

d²/dx²[arctan(x)] = -2x/(1+x²)². This shows the curvature of the arctan function.

arctan(100) ≈ 89.427° ≈ 1.5608 radians. Very close to π/2.

arctan(0.001) ≈ 0.0573° ≈ 0.001 radians. For small values, arctan(x) ≈ x.

arctan(x) = 2·arctan(x/(1+√(1+x²))) for x > 0. This allows computing arctan using simpler calculations.

Arctan appears in activation functions, attention mechanisms, and feature normalization in neural networks.

arctan(3) ≈ 71.565° or approximately 1.2490 radians.

arctan(1/3) ≈ 18.435° or approximately 0.3218 radians.

Yes, arctan(0) = 0° or 0 radians exactly.

The graph of arctan(x) is an S-shaped curve passing through the origin, with horizontal asymptotes at y = π/2 and y = -π/2.

arctan(√3/3) = 30° or π/6 radians ≈ 0.5236. Since √3/3 = 1/√3 = tan(30°).

arctan(0.8) ≈ 38.660° or approximately 0.6747 radians.

arctan(1.5) ≈ 56.310° or approximately 0.9828 radians.

The Padé approximant gives better rational approximations to arctan than Taylor series, especially near the edges of convergence.

arctan(x) = (i/2)[ln(1-ix) - ln(1+ix)] for complex x. For real x, arctan(x) = (1/2i)·ln((1+ix)/(1-ix)).

arctan(-√3) = -60° or -π/3 radians ≈ -1.0472.

arctan(-1/√3) = -30° or -π/6 radians ≈ -0.5236.

arctan(0.25) ≈ 14.036° or approximately 0.2450 radians.

arctan(4) ≈ 75.964° or approximately 1.3258 radians.

Arctan computes phase angle from real and imaginary components of a signal. Essential for Fourier analysis and filter design.

arctan(5) ≈ 78.690° or approximately 1.3734 radians.

arctan(0.2) ≈ 11.310° or approximately 0.1974 radians.

For very large x, arctan(x) approaches π/2 asymptotically. The calculator handles this gracefully without overflow.

If y = arctan(x), then x = tan(y). This reciprocal relationship defines the inverse tangent function.

arctan(1000) ≈ 89.943° ≈ 1.5698 radians. Extremely close to π/2.

arctan(0.01) ≈ 0.573° ≈ 0.01 radians. For very small values, arctan(x) ≈ x.

Arctan calculates joint angles from coordinate positions. Essential for inverse kinematics in robotic arm control.

arctan(20) ≈ 87.138° ≈ 1.5208 radians.

arctan(0.05) ≈ 2.862° ≈ 0.0500 radians.

arctan(7) ≈ 81.870° ≈ 1.4289 radians.

arctan(0.125) ≈ 7.125° ≈ 0.1244 radians.

Arctan computes angles in projectile motion, phase differences in AC circuits, and force decomposition in mechanics.

arctan(50) ≈ 88.854° ≈ 1.5508 radians.

arctan(0.02) ≈ 1.146° ≈ 0.0200 radians.

arctan(y/x) gives the angle a line from the origin to point (x,y) makes with the positive x-axis (with quadrant adjustment).

arctan(-10) ≈ -84.289° ≈ -1.4711 radians.

arctan(0.3) ≈ 16.699° ≈ 0.2915 radians.

Arctan computes bearing angles from coordinate differences. Used in GPS systems and maritime navigation calculations.

arctan(10000) ≈ 89.994° ≈ 1.5707 radians. Nearly indistinguishable from π/2.

arctan(-0.5) ≈ -26.565° ≈ -0.4636 radians.

arctan(100000) ≈ 89.9994° ≈ 1.5708 radians. Essentially π/2 in floating point.

Results show the angle in both degrees and radians, along with a step-by-step explanation of the inverse tangent computation.

arctan(0.7) ≈ 35.000° ≈ 0.6107 radians.

Arctan calculates elevation angles from horizontal distance and height difference. Essential for terrain mapping and construction.

arctan(1000000) ≈ 89.999994° ≈ 1.5708 radians. Essentially π/2 in double precision.

arctan(0.0001) ≈ 0.00573° ≈ 0.0001 radians.

arctan(x) = arctanh(ix)/i. The inverse tangent and inverse hyperbolic tangent are related through complex numbers.

arctan(0.9) ≈ 41.987° ≈ 0.7328 radians.

arctan(-2) ≈ -63.435° ≈ -1.1071 radians.

arctan(1.2) ≈ 50.194° ≈ 0.8761 radians.

Arctan computes rotation angles for objects, camera orientations, and light direction calculations in 3D rendering.

arctan(0.4) ≈ 21.801° ≈ 0.3805 radians.

arctan(8) ≈ 82.875° ≈ 1.4464 radians.

arctan(0.6) ≈ 30.964° ≈ 0.5404 radians.

For small angles (<15°), arctan(x) ≈ x in radians. For larger angles, the difference becomes significant and arctan must be used.

arctan(10000000) ≈ 89.99999994° ≈ 1.5708 radians.

arctan(x) ≈ x(6 + x²)/(6 + 4x²) is a [3,2] Padé approximant that gives better accuracy than Taylor series for |x| up to about 2.

Arctan calculates angular separations between celestial objects and altitude angles for telescope positioning.

arctan(0.35) ≈ 19.290° ≈ 0.3367 radians.

arctan(12) ≈ 85.236° ≈ 1.4876 radians.

arctan(0.45) ≈ 24.228° ≈ 0.4228 radians.

Future calculators will integrate AI for problem recognition, step-by-step solutions, voice input, and AR/VR visualizations of angle relationships.

arctan(-100) ≈ -89.427° ≈ -1.5608 radians.

arctan(0.15) ≈ 8.531° ≈ 0.1489 radians.

arctan(6) ≈ 80.538° ≈ 1.4056 radians.

arctan(0.55) ≈ 28.811° ≈ 0.5029 radians.

arctan(-0.1) ≈ -5.711° ≈ -0.0997 radians.

arctan(0.85) ≈ 40.365° ≈ 0.7045 radians.

arctan(15) ≈ 86.186° ≈ 1.5042 radians.

arctan(0.08) ≈ 4.574° ≈ 0.0798 radians.

arctan(25) ≈ 87.709° ≈ 1.5308 radians.

arctan(0.03) ≈ 1.718° ≈ 0.0300 radians.

arctan(30) ≈ 88.091° ≈ 1.5375 radians.

arctan(0.04) ≈ 2.291° ≈ 0.0400 radians.

The Arctan Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Arctan Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.