Arctan Calculator

Calculate inverse tangent (arctan) values with precision for any real number

Enter any real number to calculate its inverse tangent. The arctan function returns the angle whose tangent is the given value, with results in radians or degrees.

The domain of arctan is all real numbers (-∞ to +∞)

Examples

Click on any example to load it into the calculator

Basic Angle: arctan(0)

degrees

Calculate arctan of zero - fundamental trigonometric value

Input: 0

Unit: degrees

45° Angle: arctan(1)

degrees

Classic trigonometric value - tangent of 45° equals 1

Input: 1

Unit: degrees

60° Angle: arctan(√3)

radians

Calculate arctan of square root of 3 in radians

Input: 1.732050808

Unit: radians

Negative Value: arctan(-1)

degrees

Inverse tangent of negative one - demonstrates negative angle result

Input: -1

Unit: degrees

Other Titles
Understanding Arctan Calculator: A Comprehensive Guide
Master the inverse tangent function and its applications in trigonometry, engineering, physics, and mathematical analysis

What is Arctan? Mathematical Foundation and Concepts

  • Arctan represents the inverse of the tangent function
  • It finds the angle whose tangent equals a given value
  • Essential function in trigonometry, geometry, and engineering calculations
The arctan function, also written as tan⁻¹ or atan, is the inverse of the tangent function. It takes a real number as input and returns the angle whose tangent equals that number. This makes it invaluable for converting ratios back to angles in trigonometric calculations.
Mathematically, if tan(θ) = x, then arctan(x) = θ. For example, since tan(45°) = 1, we know that arctan(1) = 45°. This inverse relationship forms the foundation for solving countless geometric and trigonometric problems.
Unlike arcsin and arccos which have limited domains, arctan accepts any real number as input since the tangent function can produce any real value. The output, however, is restricted to the principal value range: (-π/2, π/2) radians or (-90°, 90°).
The function exhibits asymptotic behavior, approaching π/2 (90°) as the input approaches positive infinity and -π/2 (-90°) as the input approaches negative infinity. This reflects the vertical asymptotes of the tangent function.

Fundamental Arctan Values

  • arctan(0) = 0° - tangent of 0° is 0
  • arctan(1) = 45° - tangent of 45° is 1
  • arctan(√3) ≈ 60° - tangent of 60° is √3
  • arctan(-1) = -45° - demonstrates negative angle output
  • arctan(∞) = 90° - limit behavior at infinity

Step-by-Step Guide to Using the Arctan Calculator

  • Master input techniques for accurate calculations
  • Understand unit conversion between radians and degrees
  • Interpret results and apply them to real problems
Our arctan calculator provides precise calculations for any real number input with professional-grade accuracy and user-friendly interface design.
Input Guidelines:
  • No Domain Restrictions: Enter any real number. Unlike other inverse trigonometric functions, arctan accepts all real values from negative infinity to positive infinity.
  • Decimal Precision: Use high-precision decimal values for accurate calculations. The calculator handles up to 15 significant digits for maximum precision.
  • Large Values: For very large inputs (positive or negative), observe how arctan approaches its asymptotic limits of ±90° (±π/2 radians).
Unit Selection:
  • Radians: Mathematical standard unit, useful for calculus and advanced mathematics. Range: (-π/2, π/2) ≈ (-1.5708, 1.5708).
  • Degrees: Intuitive unit for most practical applications. Range: (-90°, 90°). Easier to visualize and commonly used in engineering.
Result Interpretation:
  • Principal Values: Results are always given as principal values within the specified range, ensuring unique solutions.
  • High Precision: Results display up to 6 decimal places for accuracy in engineering and scientific calculations.

Usage Examples

  • To find angle from slope 0.5: Enter 0.5, select degrees. Result: ≈ 26.57°
  • For vector angle calculation: arctan(y/x) gives direction angle
  • Converting rise/run to angle: arctan(rise/run) in construction
  • Verifying calculator accuracy: arctan(√3) should equal 60° exactly

Real-World Applications of Arctan in Engineering and Science

  • Engineering and Construction: Slope and angle calculations
  • Physics and Mechanics: Vector analysis and force decomposition
  • Computer Graphics: Rotation and orientation calculations
  • Navigation and Surveying: Bearing and direction computations
The arctan function serves critical roles across numerous fields, converting numerical ratios into meaningful angles for practical problem-solving:
Engineering and Construction:
  • Slope Analysis: Converting rise-over-run ratios to angles for ramps, roads, roofs, and structural elements. Essential for ADA compliance and safety regulations.
  • Structural Design: Determining optimal angles for trusses, braces, and support members based on load distribution requirements.
Physics and Mechanics:
  • Vector Analysis: Converting Cartesian coordinates (x,y) to polar coordinates by calculating θ = arctan(y/x) for vector direction.
  • Projectile Motion: Determining launch angles from horizontal and vertical velocity components in ballistics and sports analysis.
Computer Graphics and Gaming:
  • 2D Rotations: Calculating rotation angles for sprites, objects, and user interface elements based on mouse or touch input coordinates.
  • AI Pathfinding: Determining direction angles for character movement and object orientation in game environments.
Navigation and Surveying:
  • GPS Systems: Converting coordinate differences to bearing angles for navigation and route calculation in mapping applications.
  • Surveying: Determining property boundaries and elevation angles from horizontal and vertical distance measurements.

Real-World Applications

  • Wheelchair ramp: 1:12 slope = arctan(1/12) ≈ 4.8° (ADA compliant)
  • Vector direction: Point (3,4) has direction arctan(4/3) ≈ 53.1°
  • Road grade: 6% grade = arctan(0.06) ≈ 3.4° angle
  • GPS bearing: 100m east, 173m north → arctan(173/100) ≈ 60° bearing

Common Misconceptions and Pitfalls in Arctan Calculations

  • Understanding the restricted range of principal values
  • Avoiding confusion with atan2 for full-circle angles
  • Recognizing asymptotic behavior and computational limits
Understanding common pitfalls helps ensure accurate application of arctan in practical problems and mathematical analysis:
Principal Value Limitation:
  • Range Restriction: Arctan only returns angles between -90° and 90°. For full-circle applications, consider using atan2(y,x) which accounts for quadrant.
  • Quadrant Ambiguity: arctan(1) and arctan(-1) give 45° and -45°, but vectors (1,1) and (-1,-1) point in different directions.
Input Interpretation:
  • Units Confusion: Ensure input is the ratio value, not an angle. Common error: entering 45 expecting 45° when arctan(45) ≈ 88.7°.
  • Division by Zero: When calculating arctan(y/x), handle x=0 cases separately to avoid undefined ratios.
Computational Considerations:
  • Precision Limits: For extremely large inputs, computational precision may affect results near the asymptotic limits.
  • Degree vs Radian: Always verify the expected output unit. Many scientific calculations require radians while practical applications often use degrees.

Common Mistakes to Avoid

  • Correct: slope 0.5 → arctan(0.5) ≈ 26.57°
  • Incorrect: angle 30° → arctan(30) ≈ 88.1° (should use sin, cos, or direct conversion)
  • Vector (3,4): direction = arctan(4/3), not arctan(3/4)
  • Full circle: use atan2(y,x) for angles from -180° to 180°

Mathematical Properties and Advanced Arctan Concepts

  • Derivative and integral properties of arctan function
  • Series expansions and approximation methods
  • Relationship with other inverse trigonometric functions
Advanced mathematical properties of arctan provide deeper insight into its behavior and enable sophisticated applications:
Calculus Properties:
  • Derivative: d/dx[arctan(x)] = 1/(1+x²). This derivative is always positive, confirming arctan is strictly increasing.
  • Integral: ∫arctan(x)dx = x·arctan(x) - ½ln(1+x²) + C. Useful in advanced integration techniques.
Series Representation:
  • Taylor Series: arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ... for |x| ≤ 1. Enables numerical computation and approximation.
  • Machin's Formula: π/4 = 4·arctan(1/5) - arctan(1/239). Historical formula for calculating π with high precision.
Special Identities:
  • Addition Formula: arctan(a) + arctan(b) = arctan((a+b)/(1-ab)) when ab < 1.
  • Complementary Angles: arctan(x) + arctan(1/x) = π/2 for x > 0. Demonstrates relationship with arctangent reciprocals.
Numerical Methods:
  • CORDIC Algorithm: Hardware implementation method using iterative rotations for fast arctan computation in processors.
  • Rational Approximations: Padé approximants provide high-accuracy polynomial approximations for computational efficiency.

Advanced Mathematical Examples

  • Derivative application: maximum slope of arctan occurs at x=0 with slope=1
  • Series approximation: arctan(0.5) ≈ 0.5 - 0.125/3 + 0.03125/5 ≈ 0.4636
  • Identity verification: arctan(2) + arctan(0.5) = π/2 ≈ 1.5708 radians
  • Machin's formula: Calculate π using arctan of simple fractions