Arctan Calculator - Inverse Tangent Calculator Online

What is Arctangent (Arctan)?

The arctangent, denoted as arctan(x) or tan⁻¹(x), is the inverse function of the tangent. It returns the angle whose tangent is the given number. The arctangent function is fundamental in trigonometry and is widely used in mathematics, physics, engineering, and computer science.

Mathematical Definition

For any real number x, the arctangent is defined as:

arctan(x) = tan⁻¹(x)

The arctangent function returns values in the range (-π/2, π/2) for radians or (-90°, 90°) for degrees. This makes it a one-to-one function, unlike the tangent function which is periodic.

Key Properties

Domain: All real numbers (-∞, ∞)
Range: (-π/2, π/2) radians or (-90°, 90°) degrees
Odd function: arctan(-x) = -arctan(x)
Continuous and differentiable everywhere
Monotonic: Strictly increasing

Common Values

x	arctan(x) in Radians	arctan(x) in Degrees
0	0	0°
1	π/4	45°
√3	π/3	60°
1/√3	π/6	30°
-1	-π/4	-45°

Applications

1. Trigonometry and Geometry

Arctangent is essential for solving triangles when you know the ratio of opposite to adjacent sides and need to find the angle.

2. Physics

Used in projectile motion, wave analysis, and calculating angles in various physical systems.

3. Engineering

Critical in control systems, signal processing, and calculating phase angles in electrical engineering.

4. Computer Graphics

Used for calculating angles in 2D and 3D graphics, especially for rotations and transformations.

5. Navigation

Essential in GPS systems and navigation algorithms for calculating bearings and directions.

How to Use the Arctan Calculator

Enter a number: Input any real number in the input field
Choose output unit: Select between radians or degrees
Get instant results: The calculator provides real-time calculation
Copy results: Use the copy button to copy the result to clipboard

Examples

Example 1: Basic Calculation

Input: 1

Output in radians: 0.7853981634 rad

Output in degrees: 45°

Example 2: Negative Input

Input: -1

Output in radians: -0.7853981634 rad

Output in degrees: -45°

Example 3: Large Number

Input: 1000

Output in radians: 1.569796327 rad

Output in degrees: 89.9427°

Mathematical Relationships

The arctangent function has several important relationships:

Inverse relationship: tan(arctan(x)) = x
Derivative: d/dx[arctan(x)] = 1/(1 + x²)
Integral: ∫arctan(x)dx = x·arctan(x) - ½ln(1 + x²) + C
Series expansion: arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...

Frequently Asked Questions

What is the difference between arctan and tan?

Tan (tangent) takes an angle and returns a ratio, while arctan (arctangent) takes a ratio and returns an angle. They are inverse functions of each other.

What is the range of arctangent?

The arctangent function returns values between -π/2 and π/2 radians (or -90° and 90° degrees). This range ensures the function is one-to-one.

Can arctangent handle negative numbers?

Yes, arctangent can handle any real number, including negative numbers. For negative inputs, it returns negative angles within the function's range.

Why is arctangent important in programming?

Arctangent is crucial in programming for calculating angles, especially in graphics, game development, robotics, and navigation systems. It's commonly used to find the angle between two points or vectors.

How accurate is the arctangent calculation?

Our calculator provides results with 10 decimal places of precision, which is sufficient for most practical applications. The underlying JavaScript Math.atan() function uses high-precision algorithms.

What happens when I input very large numbers?

For very large numbers, arctangent approaches π/2 (90°) but never reaches it. The function is bounded, so extremely large inputs will give results very close to 90° but never exactly 90°.