Understanding the Arccosine Function

What is the Arccosine Function?

The arccosine function, denoted as arccos(x) or cos⁻¹(x), is the inverse function of cosine. It returns the angle (in radians) whose cosine is the input value. The arccosine function is essential in trigonometry and has numerous applications in physics, engineering, and computer graphics.

Formula and Definition

The arccosine function is defined as:

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

This means:

• If \(y = \arccos(x)\), then \(\cos(y) = x\)
• The domain of arccos(x) is [-1, 1]
• The range of arccos(x) is [0, π] or [0°, 180°]

Calculation Steps

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

1. Input: x = 0.5 (which is between -1 and 1)
2. y = arccos(0.5) ≈ 1.0472 radians
3. Converting to degrees: 1.0472 × (180/π) ≈ 60°

Visual Representation

This diagram illustrates arccos(0.5) on the unit circle. The green line represents the cosine value (0.5), and the yellow arc shows the measure of the angle (60°).