Arctan Calculator (Inverse Tangent)
Calculate inverse tangent (arctangent) values in degrees and radians with step-by-step explanations
Calculate Inverse Tangent
Domain: All real numbers (-∞, +∞)
Range: (-90°, 90°) or (-π/2, π/2) rad
Calculation Results
✓ Special angle recognized: arctan(0) = 0°
Calculation: arctan(0) = 0.000000°
Exact values: Degrees: 0.000000°, Radians: 0.000000 rad
Range: Arctan output is always between -90° and 90° (-π/2 and π/2 radians)
Related Trigonometric Values
Where θ = arctan(0) = 0.000000°
Common Arctan Values
| Input (x) | arctan(x) Degrees | arctan(x) Radians | Exact Value |
|---|---|---|---|
| 0 | 0° | 0 | 0 |
| √3/3 ≈ 0.577 | 30° | π/6 ≈ 0.524 | π/6 |
| 1 | 45° | π/4 ≈ 0.785 | π/4 |
| √3 ≈ 1.732 | 60° | π/3 ≈ 1.047 | π/3 |
| -√3/3 ≈ -0.577 | -30° | -π/6 ≈ -0.524 | -π/6 |
| -1 | -45° | -π/4 ≈ -0.785 | -π/4 |
| -√3 ≈ -1.732 | -60° | -π/3 ≈ -1.047 | -π/3 |
Arctan Properties
Domain: All real numbers (-∞, +∞)
Range: (-π/2, π/2) or (-90°, 90°)
arctan(-x) = -arctan(x) (odd function)
tan(arctan(x)) = x for all x
As x → ±∞, arctan(x) → ±π/2
Key Formulas
Basic Definition
If tan(θ) = x, then θ = arctan(x)
Derivative
d/dx[arctan(x)] = 1/(1 + x²)
Integral
∫arctan(x)dx = x·arctan(x) - ½ln(1+x²) + C
Addition Formula
arctan(x) + arctan(y) = arctan((x+y)/(1-xy))
Quick Tips
arctan(1) = 45° = π/4 radians
arctan(0) = 0° = 0 radians
arctan is the inverse of the tan function
Used to find angles in right triangles
Understanding Inverse Tangent (Arctan)
What is Arctan?
The arctangent (arctan) is the inverse function of the tangent. If tan(θ) = x, then θ = arctan(x). It answers the question: "What angle has a tangent equal to this value?"
Domain and Range
- •Domain: All real numbers (-∞, +∞)
- •Range: (-π/2, π/2) or (-90°, 90°)
- •Unlike tangent, arctan is defined for all real numbers
Mathematical Properties
Odd Function
arctan(-x) = -arctan(x)
Horizontal Asymptotes
lim(x→+∞) arctan(x) = π/2
lim(x→-∞) arctan(x) = -π/2
Monotonicity
Strictly increasing function
Practical Applications
Navigation
Calculate bearing angles from coordinate differences
Engineering
Find angles of inclination, slopes, and gradients
Physics
Calculate angles in vector problems and projectile motion