ACsc - Arccosecant (Inverse Cosecant)
Online calculator for calculating the arc cosecant (inverse cosecant)
Arccosecant Calculator
Instructions
Enter the value of the cosecant (≤-1 or ≥1), select the unit of measure (degrees or radians) and click Calculate.
Arccosecant - Overview
Value Range
The arccosecant is only defined for certain values.
- Domain: x ≤ -1 or x ≥ 1
- Range: [-π/2, π/2] excluding 0, or [-90°, 90°] excluding 0°
Inverse cosecant, scale in radians
Definition
The arccosecant (arccsc, acsc, or csc⁻¹) is the inverse function of the cosecant function.
\(\displaystyle y = \text{arccsc}(x) \Leftrightarrow \csc(y) = x \)
with \( |x| \geq 1 \) and \( y \in [-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\} \)
Relationship
Relationship to arcsine:
\(\displaystyle \text{arccsc}(x) = \arcsin\left(\frac{1}{x}\right) \)
Description of the Arccosecant
Fundamentals
The arccosecant (arccsc, acsc, or csc⁻¹) is the inverse function of the cosecant function. It calculates the angle for a given cosecant value. The cosecant is the reciprocal of the sine function.
Mathematical Definition:
\(\displaystyle y = \text{arccsc}(x) \)
means
\(\displaystyle \csc(y) = x \)
Understanding Cosecant
The cosecant is a trigonometric function that is the reciprocal of the sine function. In a right triangle:
\(\displaystyle \csc(\alpha) = \frac{1}{\sin(\alpha)} = \frac{\text{Hypotenuse}}{\text{Opposite side}} \)
Important: The cosecant is undefined when sin(α) = 0, which occurs at 0°, 180°, 360°, etc. (or 0, π, 2π, etc. in radians).
Detailed Examples
Example 1: Calculate Cosecant Value
Given:
An angle α = 30° (or π/6 radians)
Calculating the cosecant:
\(\displaystyle \csc(30°) = \frac{1}{\sin(30°)} = \frac{1}{0.5} = 2 \)
Example 2: Calculate Angle
Task:
Calculate the angle α for \( \csc(\alpha) = 2 \)
In radians:
\(\displaystyle \alpha = \text{arccsc}(2) = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6} \approx 0.524 \text{ rad} \)
Conversion to degrees:
\(\displaystyle \alpha \approx 30° \)
Example 3: Using Triangle
Given:
A right triangle with:
- Hypotenuse: c = 10
- Opposite side: a = 6
Calculation:
\(\displaystyle \csc(\alpha) = \frac{10}{6} \approx 1.667 \)
\(\displaystyle \alpha = \text{arccsc}(1.667) \approx 36.87° \)
Conversion Formula
From radians to degrees:
\(\displaystyle \text{Degrees} = \frac{\text{Radians} \cdot 180°}{\pi} \)
Properties
- Domain: \( x \leq -1 \) or \( x \geq 1 \)
- Range: \( y \in [-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\} \) (radians) or \( [-90°, 90°] \setminus \{0°\} \)
- Discontinuity: Undefined at x = 0
- Reciprocal: \( \csc(x) = \frac{1}{\sin(x)} \)
- Special values:
- \( \text{arccsc}(1) = \frac{\pi}{2} = 90° \)
- \( \text{arccsc}(2) = \frac{\pi}{6} = 30° \)
- \( \text{arccsc}(-1) = -\frac{\pi}{2} = -90° \)
- \( \text{arccsc}(\sqrt{2}) = \frac{\pi}{4} = 45° \)
Practical Applications
- Physics: Wave mechanics and oscillations
- Engineering: Structural analysis
- Astronomy: Calculating celestial angles
- Navigation: Position and direction calculations
- Optics: Light refraction calculations
- Signal processing: Amplitude analysis
- Computer graphics: 3D rendering and projections
Relationships to Other Functions
Sine relationship:
\(\displaystyle \csc(\alpha) = \frac{1}{\sin(\alpha)} \)
\(\displaystyle \sin(\alpha) = \frac{1}{\csc(\alpha)} \)
Arcsine relationship:
\(\displaystyle \text{arccsc}(x) = \arcsin\left(\frac{1}{x}\right) \)
for \( |x| \geq 1 \)
Pythagorean Identity
\(\displaystyle 1 + \cot^2(\alpha) = \csc^2(\alpha) \)
This identity relates the cosecant to the cotangent
Important Note
Domain restriction: The arccosecant is only defined for values where \( |x| \geq 1 \). For values between -1 and 1, the function is undefined because the sine function (whose reciprocal is the cosecant) only produces values in the range [-1, 1].
Programming: Many programming languages do not have a
built-in arccosecant function. It can be calculated using:
acsc(x) = asin(1/x).