Tan⁻¹ Calculator
Calculate inverse tangent (arctan) values in radians and degrees with step-by-step explanations
Calculate Tan⁻¹ (Inverse Tangent)
Any real number (domain: all real numbers)
Choose your preferred angle unit
Tan⁻¹ Results
Function: tan⁻¹(0) = arctan(0)
Calculation: Using the inverse tangent function
Exact Value: tan⁻¹(0) = 0
Property: tan⁻¹(-x) = -tan⁻¹(x) (odd function)
Function Properties
Common Values
Special Values
tan⁻¹(0) = 0 rad = 0°
tan⁻¹(1) = π/4 rad = 45°
tan⁻¹(-1) = -π/4 rad = -45°
tan⁻¹(√3) = π/3 rad = 60°
tan⁻¹(1/√3) = π/6 rad = 30°
Properties
Odd function: tan⁻¹(-x) = -tan⁻¹(x)
Limits: lim(x→∞) tan⁻¹(x) = π/2
Limits: lim(x→-∞) tan⁻¹(x) = -π/2
Derivative: d/dx[tan⁻¹(x)] = 1/(1+x²)
Notation Guide
tan⁻¹(x)
Inverse tangent function
Most common notation
arctan(x)
Arc tangent function
Preferred mathematical notation
atan(x)
Programming notation
Common in code and calculators
Unit Circle Reference
Quadrant I
0 < tan⁻¹(x) < π/2
For x > 0
Quadrant IV
-π/2 < tan⁻¹(x) < 0
For x < 0
Principal Range
(-π/2, π/2) radians
(-90°, 90°)
Quick Tips
tan⁻¹(x) accepts any real number as input
Output is always between -90° and 90°
tan⁻¹(1) = 45° is a key reference angle
The function is increasing for all x
Understanding Tan⁻¹ (Inverse Tangent)
What is Tan⁻¹?
The inverse tangent function, denoted as tan⁻¹(x) or arctan(x), is the inverse of the tangent function. It answers the question: "What angle has a tangent equal to x?" The result is always given in the principal range of (-90°, 90°) or (-π/2, π/2) radians.
Important Properties
- •Domain: All real numbers (-∞, +∞)
- •Range: (-π/2, π/2) or (-90°, 90°)
- •Odd Function: tan⁻¹(-x) = -tan⁻¹(x)
- •Monotonic: Always increasing function
Notation Confusion
Important: The notation tan⁻¹(x) can be confusing! It typically means the inverse tangent function (arctan), NOT 1/tan(x) which is cotangent.
Common Applications
- Trigonometry: Finding angles from ratios
- Physics: Calculating angles of incidence
- Engineering: Slope angle calculations
- Navigation: Bearing and heading calculations
- Computer Graphics: Rotation calculations
Memory Tip: Remember that tan⁻¹(1) = 45° because tan(45°) = 1