Arctan Calculator | Inverse Tangent Calculator
Calculate arctan/inverse tangent values with precision. Convert between degrees/radians, visualize angles, and use atan2 for coordinates.
The Arctan (Inverse Tangent) Calculator is a mathematical tool that helps you find the angle whose tangent is a given number. Arctan is the inverse function of the tangent trigonometric function, widely used in mathematics, physics, engineering, and computer graphics for angle calculations and coordinate transformations.
What is Arctan?
Arctan, also known as inverse tangent or tan⁻¹, is the inverse trigonometric function of tangent. For a given real number x, arctan(x) returns the angle θ (in radians or degrees) such that tan(θ) = x. The range of arctan is typically (-π/2, π/2) or (-90°, 90°), making it particularly useful for determining angles in right triangles and coordinate systems.
Mathematical Definition
Key Features
- Dual Output Units: Calculate results in both radians and degrees simultaneously.
- Visual Angle Representation: See a visual representation of the calculated angle in a unit circle.
- Multiple Input Formats: Enter values as decimals, fractions, or trigonometric expressions.
- Coordinate Input: Calculate angle from (x, y) coordinates using atan2 function.
- Precision Control: Adjust decimal places from 2 to 15 for precise calculations.
- History Tracking: Save and compare multiple calculations with timestamps.
- Graph Visualization: View the arctan function graph around your input value.
- Mobile Responsive: Works perfectly on all devices including desktops, tablets, and smartphones.
Applications of Arctan
Trigonometry & Geometry
Find angles in right triangles when opposite and adjacent sides are known.
Physics & Engineering
Calculate launch angles, slopes, and direction vectors in mechanics.
Computer Graphics
Determine rotation angles for 2D and 3D transformations.
Navigation & GIS
Calculate bearings and directions from coordinate differences.
Common Arctan Values
| x (tan θ) | Angle (Degrees) | Angle (Radians) | Common Application |
|---|---|---|---|
| 0 | 0° | 0 | Horizontal line |
| 1/√3 ≈ 0.5774 | 30° | π/6 ≈ 0.5236 | Standard angle |
| 1 | 45° | π/4 ≈ 0.7854 | Equal rise/run |
| √3 ≈ 1.7321 | 60° | π/3 ≈ 1.0472 | Standard angle |
| ∞ | 90° | π/2 ≈ 1.5708 | Vertical line (limit) |
Arctan vs Atan2
Arctan(x) Function
- Single input: ratio y/x
- Range: (-π/2, π/2) or (-90°, 90°)
- Undefined for x = 0
- Used when quadrant is known
- Simple ratio calculations
Atan2(y, x) Function
- Two inputs: y and x coordinates
- Range: (-π, π] or (-180°, 180°]
- Handles all quadrants
- Determines correct quadrant
- Used in navigation and graphics
Calculation Methods
Taylor Series Expansion
arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ... for |x| ≤ 1. This infinite series converges for values between -1 and 1.
CORDIC Algorithm
Uses iterative shift-and-add operations to calculate trigonometric functions efficiently in digital systems.
Lookup Tables
Precomputed values for common inputs with interpolation for intermediate values in embedded systems.
Important Considerations
- Arctan returns principal values only (between -90° and 90°)
- For angles in other quadrants, use atan2(y, x) function
- arctan(∞) approaches 90° but is mathematically undefined at infinity
- Be mindful of units (radians vs degrees) in your calculations
- Inverse trigonometric functions have restricted domains
- Check for numerical stability with very large or very small inputs
Frequently Asked Questions
What's the difference between arctan and tan⁻¹?
They are the same function. arctan(x) and tan⁻¹(x) both represent the inverse tangent function. The "arc" prefix refers to the arc length on the unit circle, while the superscript "-1" denotes the inverse function.
Why does arctan only return angles between -90° and 90°?
This is the principal value range where the tangent function is one-to-one and therefore invertible. For angles outside this range, the tangent function repeats (periodic), so restricting the range ensures a unique inverse.
When should I use atan2 instead of arctan?
Use atan2(y, x) when you have separate x and y coordinates and need to determine the correct quadrant of the angle. Use arctan(y/x) when you only have the ratio and already know which quadrant the angle should be in.
How accurate is the arctan calculation?
Modern calculators and programming languages typically compute arctan to 15-16 decimal places of accuracy. Our calculator provides adjustable precision from 2 to 15 decimal places based on your needs.
This arctan calculator is intended for educational, scientific, and engineering purposes. The calculations are based on standard mathematical functions and algorithms. For critical applications requiring extreme precision, verify results with specialized mathematical software. Always consider the context and units when applying inverse trigonometric functions in real-world scenarios.