Understanding the Arctangent Function

What is the Arctangent Function?

The arctangent function, denoted as arctan(x) or tan⁻¹(x), is the inverse function of tangent. It returns the angle (in radians) whose tangent is the input value. The arctangent function is crucial in trigonometry and has numerous applications in physics, engineering, and computer graphics.

Formula and Definition

The arctangent function is defined as:

\[y = \arctan(x) \quad \text{for all real } x\]

This means:

• If \(y = \arctan(x)\), then \(\tan(y) = x\)
• The domain of arctan(x) is all real numbers
• The range of arctan(x) is (-π/2, π/2) or (-90°, 90°)

Calculation Steps

1. Input any real number x.
2. Apply the arctangent function: y = arctan(x).
3. The result y will be in radians.
4. If needed, convert the result to degrees using the formula: degrees = radians × (180/π).

Example Calculation

Let's calculate arctan(1):

1. Input: x = 1
2. y = arctan(1) ≈ 0.7854 radians
3. Converting to degrees: 0.7854 × (180/π) ≈ 45°

Visual Representation

This diagram illustrates arctan(1) on the coordinate plane. The green line represents the tangent value (1), and the yellow arc shows the measure of the angle (45°).