Inverse Tangent Calculator
Calculate arctan(x) with precise results in radians and degrees. Our calculator recognizes special values, provides comprehensive analysis, and shows step-by-step solutions for inverse tangent calculations with quadrant analysis and related angle computations for complete trigonometric understanding.
Last updated: December 15, 2024
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Enter any real number. Common values: 0, ±1, ±√3, ±1/√3
Inverse Tangent Result
Input:
Results:
Special Value:
Quadrant Analysis:
Related Angles:
Alternative Formats:
Calculation Steps:
- Step 2: Calculate arctan(1)
- Step 3: arctan(1) = 0.785398 radians
- Step 4: Convert to degrees: 0.785398 × (180/π) = 45.000000°
- Step 5: This is a special value: π/4
- Step 6: Principal value is in quadrant 1
- Step 7: Final result: 0.785398 radians = 45.000000°
Quick Example Result
For input value 1:
arctan(1) = 0.785398 rad = 45.000000°
How This Calculator Works
Our inverse tangent calculator uses advanced trigonometric algorithms to compute arctan(x) with high precision. The calculator recognizes special values, performs quadrant analysis, and provides comprehensive results in multiple formats. It handles both decimal and fractional inputs with detailed step-by-step explanations for educational and professional applications.
Inverse Tangent Calculation Process
Input Processing:
Convert fractions/expressions to decimal formPrincipal Value Calculation:
θ = arctan(x), where -π/2 < θ < π/2Unit Conversion:
degrees = radians × (180/π)Shows the arctan function curve with asymptotes at ±π/2
Mathematical Foundation
The inverse tangent function is defined as the inverse of the tangent function over its principal domain. For any real number x, arctan(x) returns the unique angle θ in the interval (-π/2, π/2) such that tan(θ) = x. This function is continuous and strictly increasing over its entire domain, with horizontal asymptotes at y = ±π/2 as x approaches ±∞.
- Domain: All real numbers (-∞, ∞)
- Range: (-π/2, π/2) or (-90°, 90°)
- Special values: arctan(0) = 0, arctan(1) = π/4, arctan(√3) = π/3
- Asymptotes: Horizontal asymptotes at y = ±π/2
Sources & References
- Stewart Calculus - Inverse Trigonometric FunctionsComprehensive coverage of inverse trig functions and their properties
- Wolfram MathWorld - Inverse Tangent FunctionMathematical definitions, properties, and special values
- Khan Academy - Inverse Trigonometric FunctionsInteractive tutorials on inverse trig functions and applications
Need help with other trigonometric functions? Check out our coterminal angle calculator and angle between vectors calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Input:
- Input Value: 1
- Function: arctan(1)
- Special Value: Yes
Calculation Steps:
- Identify input: arctan(1)
- Recognize special value: tan(π/4) = 1
- Therefore: arctan(1) = π/4 radians
- Convert: π/4 × (180/π) = 45°
Result: arctan(1) = π/4 radians = 45° (Special Value)
This is one of the fundamental angles in trigonometry, representing a 45° angle where opposite and adjacent sides are equal.
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