Understanding the Arcsine Function

What is the Arcsine Function?

The arcsine function, denoted as arcsin(x) or sin⁻¹(x), is the inverse function of sine. It returns the angle (in radians) whose sine is the input value. The arcsine function is fundamental in trigonometry and has numerous applications in physics, engineering, and computer graphics.

Formula and Definition

The arcsine function is defined as:

\[y = \arcsin(x) \quad \text{where} \quad -1 \leq x \leq 1\]

This means:

• If \(y = \arcsin(x)\), then \(\sin(y) = x\)
• The domain of arcsin(x) is [-1, 1]
• The range of arcsin(x) is [-π/2, π/2] or [-90°, 90°]

Calculation Steps

1. Ensure the input value x is between -1 and 1 (inclusive).
2. Apply the arcsine function: y = arcsin(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 arcsin(0.5):

1. Input: x = 0.5 (which is between -1 and 1)
2. y = arcsin(0.5) ≈ 0.5236 radians
3. Converting to degrees: 0.5236 × (180/π) ≈ 30°

Visual Representation

This diagram illustrates arcsin(0.5) on the unit circle. The green line represents the sine value (0.5), and the yellow arc shows the measure of the angle (30°).