Arctan Calculator — How to Find tan⁻¹(x) in Degrees and Radians
What Does This Arctan Calculator Do?
This arctan calculator computes the inverse tangent of any real number and returns the angle in degrees or radians. Enter a value x, and the calculator instantly finds tan⁻¹(x) — the angle whose tangent equals x. Whether you need arctan(1) = 45°, arctan(0) = 0°, or arctan of any decimal, this tool gives you the answer with adjustable precision, step-by-step work, and derived trigonometric values.
The arctangent function is one of the most frequently used inverse trigonometric functions in engineering, physics, and mathematics. If you know the ratio of the opposite side to the adjacent side in a right triangle (the tangent), arctan gives you back the angle. This is essential for converting between rectangular and polar coordinates, finding the direction of a vector, or determining the slope angle of a line.
Our arctan calculator handles three input modes: a single value for standard arctan(x), a rise/run ratio for slope-to-angle conversion, and atan2(y, x) for full four-quadrant angle calculation. Every result includes both degrees and radians, exact values when applicable, and the corresponding sin, cos, and tan of the resulting angle.
How to Calculate Arctan — Step by Step
Calculating arctan (inverse tangent) means finding the angle θ such that tan(θ) = x. Here is how you use this arctan calculator:
- Enter your value — type any real number into the input field. The domain of arctan is all real numbers, so any value from -∞ to +∞ is valid.
- Choose your output unit — select degrees or radians depending on your application.
- Set precision — choose 0 to 8 decimal places for your result.
- Click Calculate (or press Enter) — the calculator returns arctan(x) with the exact value when it exists, both degree and radian forms, quadrant information, and step-by-step work.
For example, to find arctan(1) in degrees: enter 1, select Degrees, and the result is 45° (exactly π/4 radians). To find arctan(10) in degrees: enter 10, and you get 84.2894° — close to 90° because large values of x push arctan toward its horizontal asymptote.
Arctan Formula, Domain, and Range
The arctan formula is defined as the inverse of the tangent function restricted to its principal branch:
- Domain of arctan: All real numbers (-∞, +∞). Unlike arcsin and arccos, arctan accepts any input because the tangent function can produce any value.
- Range of arctan: (-90°, 90°) or (-π/2, π/2) in radians. This is the principal value — the unique angle returned for each input.
- Arctan is an odd function: arctan(-x) = -arctan(x). This means the graph is symmetric about the origin.
- Horizontal asymptotes: arctan(x) → 90° as x → +∞, and arctan(x) → -90° as x → -∞.
Common Arctan Values in Degrees and Radians
These are the standard arctan values you should know from the unit circle. Each corresponds to a well-known angle:
- arctan(0) = 0° = 0 radians — because tan(0°) = 0
- arctan(√3/3) = 30° = π/6 radians — because tan(30°) = √3/3 ≈ 0.5774
- arctan(1) = 45° = π/4 radians — the most frequently searched arctan value
- arctan(√3) = 60° = π/3 radians — because tan(60°) = √3 ≈ 1.7321
- arctan(10) ≈ 84.29° — demonstrating how large inputs approach 90°
- arctan(100) ≈ 89.43° — very close to the 90° asymptote but never reaching it
For negative inputs, simply negate the result: arctan(-1) = -45°, arctan(-√3) = -60°. This follows from the odd-function property of arctangent.
Arctan vs atan2 — When to Use Each
A common source of confusion is the difference between arctan(y/x) and atan2(y, x). Our calculator supports both modes:
- arctan(y/x) returns angles only in (-90°, 90°) — it cannot distinguish between quadrants II and IV, or I and III.
- atan2(y, x) takes both coordinates separately and returns angles in (-180°, 180°], correctly identifying all four quadrants.
Use atan2 when working with coordinates, vectors, or any application where direction matters. Use standard arctan when you only need the angle from a single tangent value or slope.
Converting Arctan to Degrees
Many searches ask how to convert arctan in degrees vs radians. Most programming languages and scientific calculators return arctan in radians by default. To convert arctan from radians to degrees, multiply by 180/π:
For example: arctan(1) = π/4 radians = (π/4) × (180°/π) = 45°. Our arctan calculator handles this conversion automatically — just select "Degrees" from the dropdown and your result displays in degrees with your chosen precision.
This is especially useful when you need the result of an arctan calculation in degrees for geometry problems, compass bearings, or engineering specifications that use degree notation.
Practical Applications of Arctan
Slope-to-Angle Conversion in Engineering
The most common engineering use of arctan is converting a slope (rise over run) to an angle. If a road has a 6% grade (rise/run = 0.06), the road angle is arctan(0.06) = 3.43°. For a roof with a 6:12 pitch, arctan(6/12) = 26.57°. Use the Rise/Run mode in this calculator to compute slope angles directly.
Vector Direction and Navigation
To find the direction angle of a vector (x, y), you calculate arctan(y/x) — or more precisely, atan2(y, x) for correct quadrant handling. GPS systems, game physics engines, and robotics all rely on this calculation for determining heading and orientation. Switch to the atan2 mode to compute full four-quadrant angles.
Calculus and Integration
In calculus, arctan appears as the antiderivative of 1/(1+x²): ∫ 1/(1+x²) dx = arctan(x) + C. This integral is fundamental in physics (electric potential, fluid dynamics) and probability (the Cauchy distribution). The derivative of arctan(x) is 1/(1+x²), which makes it essential in many differential equations.
Students: How to Find Arctan on a Calculator
On a scientific calculator, arctan is accessed by pressing SHIFT (or 2nd/INV) followed by TAN. Make sure your calculator is set to the correct angle mode (DEG or RAD) before computing. On this online arctan calculator, simply type your value and select degrees or radians — no mode switching needed. Pair this tool with our tan calculator for verification and our inverse tangent calculator for additional inverse trig features.
Related Trigonometric Calculators
Explore more inverse trigonometric functions and trigonometry tools:
- Arcsin Calculator — find sin⁻¹(x) for values between -1 and 1
- Arccos Calculator — find cos⁻¹(x) for values between -1 and 1
- Unit Circle Calculator — explore all trig values on the unit circle
- Trigonometry Calculator — comprehensive trig function toolkit
- Angle Calculator — convert between angle units and formats
- All Geometry & Trigonometry Calculators
About the Author
Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
Why Use This Arctan Calculator?
This free online arctan calculator gives you instant, precise results for any inverse tangent calculation. It supports three input modes (single value, slope ratio, and atan2 coordinates), outputs in both degrees and radians simultaneously, recognizes exact values from the unit circle, and shows the complete solution with step-by-step work. Whether you are a student checking homework, an engineer computing angles, or a programmer implementing trigonometric logic — bookmark this page for reliable arctan calculations.
