1. What is Cosine Inverse?

The cosine inverse function (arccos) is the inverse of the cosine function. It returns the angle whose cosine is the given number. The output range is [0, π] radians or [0°, 180°] in degrees.

2. How Does the Calculator Work?

The calculator uses the arccos function:

Where:

• \( x \) — Input value between -1 and 1
• \( \theta \) — Resulting angle in radians or degrees

Explanation: The function calculates the principal value of the angle whose cosine is x. For any real number x between -1 and 1, arccos(x) is the angle θ in the interval [0, π] radians such that cos(θ) = x.

3. Importance of Arccos Calculation

Details: The arccos function is essential in trigonometry, physics, engineering, and computer graphics for determining angles from cosine values. It's particularly useful in solving triangles and analyzing periodic phenomena.

4. Using the Calculator

Tips: Enter a value between -1 and 1 (inclusive). The calculator will return the angle in either degrees or radians based on your selection. Invalid inputs (outside [-1, 1]) will be rejected.

5. Frequently Asked Questions (FAQ)

Q1: Why is the input range limited to [-1, 1]?
A: The cosine of any real angle always falls between -1 and 1, so arccos is only defined for inputs in this range.

Q2: What's the difference between degrees and radians?
A: Degrees and radians are two units for measuring angles. 180° equals π radians. Degrees are more common in everyday applications, while radians are preferred in higher mathematics.

Q3: Can I get angles beyond 180°?
A: The principal value of arccos is always between 0° and 180° (0 and π radians). Other angles with the same cosine value can be obtained using trigonometric identities.

Q4: How precise is the calculation?
A: The calculator uses PHP's built-in acos() function which provides high precision (typically about 15 decimal digits).

Q5: What about complex numbers?
A: This calculator only handles real number inputs. For complex numbers, a more advanced calculator would be needed.