Inverse Tangent Calculator
Calculate arctan(x) - the inverse tangent function for any real number
Calculate Inverse Tangent
Domain: All real numbers (-∞, ∞)
Results
In Degrees
In Radians
Explanation
tan(45°) = 1, therefore arctan(1) = 45°
Special Values of Inverse Tangent
| Input (x) | arctan(x) in Degrees | arctan(x) in Radians | Exact Value |
|---|---|---|---|
| -√3 | -60° | -π/3 | -π/3 ≈ -1.0472 |
| -1 | -45° | -π/4 | -π/4 ≈ -0.7854 |
| -1/√3 | -30° | -π/6 | -π/6 ≈ -0.5236 |
| 0 | 0° | 0 | 0 |
| 1/√3 | 30° | π/6 | π/6 ≈ 0.5236 |
| 1 | 45° | π/4 | π/4 ≈ 0.7854 |
| √3 | 60° | π/3 | π/3 ≈ 1.0472 |
Quick Reference
Properties
Understanding the Inverse Tangent Function
What is Inverse Tangent?
The inverse tangent function, denoted as arctan(x) or tan⁻¹(x), is the inverse function of the tangent. It takes any real number as input and returns the angle whose tangent equals that value.
Key Properties
- •Domain: All real numbers (-∞, ∞)
- •Range: (-π/2, π/2) radians or (-90°, 90°)
- •The function is increasing (as x increases, arctan(x) increases)
- •arctan(x) is an odd function: arctan(-x) = -arctan(x)
- •Has horizontal asymptotes at y = ±π/2
Applications
The inverse tangent function is widely used in mathematics, engineering, physics, computer graphics, and machine learning for angle calculations and activation functions.
Common Use Cases
- •Finding angles in coordinate geometry
- •Solving trigonometric equations
- •Navigation and GPS calculations
- •Computer graphics and 3D rotations
- •Machine learning activation functions
Key Formulas
Basic Definition
Relationships
Step-by-Step Examples
Example 1: Find arctan(1)
Step 1: We need to find the angle whose tangent is 1
Step 2: From the unit circle, we know tan(45°) = 1
Step 3: Since 45° is in the range (-90°, 90°), arctan(1) = 45°
Answer: arctan(1) = 45° = π/4 radians
Example 2: Find arctan(√3)
Step 1: We need to find the angle whose tangent is √3
Step 2: From the unit circle, we know tan(60°) = √3
Step 3: Since 60° is in the range (-90°, 90°), arctan(√3) = 60°
Answer: arctan(√3) = 60° = π/3 radians
Example 3: Find arctan(0)
Step 1: We need to find the angle whose tangent is 0
Step 2: From the unit circle, we know tan(0°) = 0
Step 3: Since 0° is in the range (-90°, 90°), arctan(0) = 0°
Answer: arctan(0) = 0° = 0 radians