1. What is Arc Cosecant?

The arc cosecant (arccsc) is the inverse function of the cosecant function. It returns the angle whose cosecant is a given number. The domain of arccsc is restricted to x ≤ -1 or x ≥ 1.

2. How Does the Calculator Work?

The calculator uses the mathematical relationship:

Where:

• \( x \) — Input value (must be ≥ 1 or ≤ -1)
• \( \arcsin \) — Inverse sine function

Explanation: The calculator first computes 1/x, then finds the arcsine of that value, giving the result in both radians and degrees.

3. Mathematical Explanation

Details: The arc cosecant function is defined as the inverse of the cosecant function, restricted to its principal values. The relationship with arcsine comes from the trigonometric identity.

4. Using the Calculator

Tips: Enter any real number x where |x| ≥ 1. The calculator will return the principal value of arccsc(x) in both radians and degrees.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of arccsc(x)?
A: The range is [-π/2, 0) ∪ (0, π/2] radians or [-90°, 0°) ∪ (0°, 90°] in degrees.

Q2: Why is the domain restricted?
A: Because cosecant values only exist outside the interval (-1, 1), the inverse function is only defined there.

Q3: How is this different from arcsin?
A: Arcsin gives the angle for a given sine ratio, while arccsc gives the angle for a given cosecant ratio.

Q4: What about negative values?
A: For x ≤ -1, arccsc(x) = -arcsin(1/|x|).

Q5: What are practical applications?
A: Used in trigonometry, calculus, physics, and engineering problems involving inverse trigonometric functions.