Calculate inverse tangent (arctan) values in degrees, radians, and gradians. Find the angle θ from any tangent ratio with step-by-step results.
A wheelchair ramp rises 1 meter over a horizontal distance of 12 meters.
Formula: θ = arctan(opposite / adjacent) = arctan(1/12)
arctan(0.0833) ≈ 4.76°
This meets the ADA recommendation of a 1:12 slope ratio for wheelchair ramps.
A roof has a rise of 4 meters over a run of 6 meters.
Formula: θ = arctan(rise / run) = arctan(4/6)
arctan(0.6667) ≈ 33.69°
Roof pitch is commonly expressed as rise over run, and arctangent converts this ratio to the roof angle.
A ship travels 20 km east and 15 km north. What is the bearing from its starting point?
Formula: θ = arctan(north / east) = arctan(15/20)
arctan(0.75) ≈ 36.87° (bearing 036.9°)
Arctangent is essential in navigation to convert displacement ratios into directional bearings.
An astronomer observes a star at a parallax angle. The star's distance from Earth is 100 AU with a baseline of 1 AU.
Formula: θ = arctan(baseline / distance) = arctan(1/100)
arctan(0.01) ≈ 0.573° (34.4 arcminutes)
Arctangent is key in astronomy for calculating angular measurements from distance and baseline ratios.
The arctangent function (arctan or tan⁻¹) is the inverse of the tangent function. For a given value x, arctan(x) returns the angle θ whose tangent is x. In a right triangle, this is the angle whose opposite/adjacent ratio equals the input value.
arctan(x) always returns an angle between -90° and 90° (exclusive). For x > 0, the angle is in Quadrant I. For x < 0, it is in Quadrant IV.
arctan(-x) = -arctan(x). The arctangent is an odd function, symmetric about the origin.
As x → ∞, arctan(x) → 90° (π/2). As x → -∞, arctan(x) → -90° (-π/2). The function approaches but never reaches these limits.
90° = 100 gradians. arctan(1) = 45° = 50 gon. arctan(∞) → 90° = 100 gon. Use gradians for metric-based surveying.
The arctangent function (abbreviated arctan, tan⁻¹, or atan) is the inverse of the tangent function. While the tangent function takes an angle and returns a ratio, the arctangent function takes a ratio and returns the corresponding angle. It answers the question: "What angle θ has a tangent equal to x?"
In a right triangle, the tangent of an angle θ is the ratio of the opposite side to the adjacent side: tan(θ) = opposite/adjacent. Therefore, arctan(opposite/adjacent) = θ. This makes arctangent essential for determining angles when you know the side lengths of a triangle.
The arctangent function has a range of (-π/2, π/2) radians (or -90° to 90°), meaning it always returns a principal value within this interval. This is the range where the tangent function is one-to-one and thus invertible. Arctangent is an odd function, satisfying arctan(-x) = -arctan(x).
Calculating arctangent values can be done in several ways. The most direct method is using the inverse trigonometric functions available on scientific calculators or in programming languages (atan, Math.atan, or arctan). For common values, the arctangent can be derived from known tangent ratios.
Method 1 — Direct Calculation: Enter the tangent value x into a calculator and press the tan⁻¹ or arctan function. For example, arctan(1) = 45° because tan(45°) = 1.
Method 2 — Ratio Method: If you know the opposite and adjacent sides of a right triangle, divide the opposite by the adjacent to get x, then apply the arctan function. For example, a ramp that rises 2 meters over 5 meters has an angle of arctan(2/5) = arctan(0.4) ≈ 21.8°.
Method 3 — Special Values: For the common angles (0°, 30°, 45°, 60°), the tangent values are exact: tan(0°) = 0, tan(30°) = 1/√3 ≈ 0.5774, tan(45°) = 1, tan(60°) = √3 ≈ 1.732. The arctangent of these values gives the corresponding angle.
These exact values are the building blocks of inverse trigonometry:
The arctangent function may seem like a purely mathematical concept, but it has numerous practical applications that affect our everyday lives. Understanding arctangent helps in fields ranging from construction to robotics, from navigation to game development.
Building codes specify maximum slope ratios for ramps (1:12) and stairs (1:2). Arctangent converts these ratios to angles, helping architects and builders ensure compliance.
When you get turn-by-turn directions, arctangent (as atan2) calculates the bearing angle between your current position and the next waypoint, converting latitude/longitude differences into compass directions.
In game development, arctan2 is used to rotate a character or object to face the mouse cursor or another game object. It's also used for aiming mechanics and camera controls.
Robotic arms use inverse kinematics, heavily relying on arctangent to calculate joint angles required to reach a specific (x, y) position in space.
⚠️ Important Note: This Arctangent Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering, construction, navigation, or medical devices. Always consult a qualified professional for decisions involving trigonometric calculations in high-stakes contexts.