Cosecant Calculator - Calculate csc(x) and arccsc(x)

What is the Cosecant Function?

The cosecant function, denoted as csc(x), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function, representing the ratio of the hypotenuse to the opposite side in a right triangle.

The cosecant function is widely used in mathematics, physics, engineering, and various scientific applications. It's particularly important in calculus, where it appears in derivatives and integrals of trigonometric functions.

The mathematical definition of cosecant is:

Key properties of the cosecant function include:

• Domain: csc(x) is defined for all real numbers except x = nπ, where n is any integer.
• Range: The cosecant function has a range of (-∞, -1] ∪ [1, ∞).
• Periodicity: csc(x) is periodic with period 2π.
• Symmetry: csc(-x) = -csc(x), making it an odd function.
• Asymptotes: Vertical asymptotes occur at x = nπ.
• Derivative: The derivative of csc(x) is -csc(x)cot(x).

The cosecant function is essential in solving trigonometric equations, analyzing periodic phenomena, and in applications involving right triangles and circular motion.

What is Inverse Cosecant (Arccosecant)?

The inverse cosecant function, denoted as arccsc(x) or csc⁻¹(x), is the inverse function of the cosecant. It answers the question: 'What angle has a cosecant of x?' The inverse cosecant function returns the angle whose cosecant is the given value.

The mathematical definition of inverse cosecant is:

Properties of the inverse cosecant function:

• Domain: The inverse cosecant is defined for |x| ≥ 1.
• Range: The principal value range is [-π/2, 0) ∪ (0, π/2].
• Monotonicity: arccsc(x) is strictly decreasing on its domain.
• Special values: arccsc(1) = π/2, arccsc(2) = π/6, arccsc(√2) = π/4.
• Derivative: The derivative of arccsc(x) is -1/(|x|√(x²-1)).

The inverse cosecant function is particularly useful in solving trigonometric equations and in applications where you need to find angles from cosecant values.

Common Cosecant Values

Here are some important cosecant values for common angles:

• csc(0°) = undefined
• csc(30°) = 2
• csc(45°) = √2 ≈ 1.414
• csc(60°) = 2/√3 ≈ 1.155
• csc(90°) = 1
• csc(120°) = 2/√3 ≈ 1.155
• csc(135°) = √2 ≈ 1.414
• csc(150°) = 2