1. What is Inverse Cosine?

The inverse cosine function, denoted as arccos(x) or cos⁻¹(x), is the inverse operation of the cosine function. It returns the angle whose cosine is the given number. The output range is [0, π] radians or [0°, 180°].

2. How Does the Calculator Work?

The calculator computes the inverse cosine using the PHP acos() function, which returns the result in radians. The result is then converted to degrees using the rad2deg() function.

Where:

• \( x \) — Input value (must be between -1 and 1 inclusive)
• \( \theta \) — Resulting angle in radians/degrees

3. Mathematical Definition

Definition: For x ∈ [-1, 1], arccos(x) is the unique angle θ ∈ [0, π] such that cos(θ) = x.

Domain: [-1, 1]
Range: [0, π] radians or [0°, 180°]

4. Using the Calculator

Instructions: Enter a value between -1 and 1 in the input field. The calculator will compute both the radian and degree equivalents of the inverse cosine.

Note: The input value must be within the range [-1, 1]. Values outside this range are mathematically undefined for the inverse cosine function.

5. Frequently Asked Questions (FAQ)

Q1: Why is the input restricted to [-1, 1]?
A: The cosine function only produces outputs in the range [-1, 1], so its inverse is only defined for inputs within this range.

Q2: What is the difference between arccos and cos⁻¹?
A: Both notations represent the same inverse cosine function. arccos(x) and cos⁻¹(x) are equivalent.

Q3: Can I calculate inverse cosine for complex numbers?
A: This calculator only handles real numbers. Complex number calculations require specialized mathematical software.

Q4: How accurate are the results?
A: The results are computed using PHP's built-in mathematical functions, which provide high precision calculations.

Q5: What are some practical applications of inverse cosine?
A: Inverse cosine is used in trigonometry, physics, engineering, computer graphics, and navigation to find angles from cosine values.