About

The cosecant function csc(θ) = 1sin(θ) is undefined wherever sin(θ) = 0, which occurs at every integer multiple of π radians (0°, 180°, 360°, …). Feeding such values into a naive calculator returns Infinity or NaN without explanation. This tool performs domain validation before computation, flags undefined inputs explicitly, and reports the intermediate sin(θ) value so you can verify the reciprocal yourself. It also detects special angles and returns exact symbolic forms like 2⋅√3÷3 rather than lossy floating-point approximations.

The calculator accepts degrees, radians, and gradians. Gradians are standard in surveying and geodesy (a right angle = 100grad). Note: floating-point arithmetic means sin(180°) evaluates to approximately 1.22×10⁻¹⁶ rather than exact zero. This tool applies an epsilon threshold of 10⁻¹² to correctly identify these near-zero cases as undefined.

Formulas

The cosecant is defined as the reciprocal of the sine function:

csc(θ) = 1sin(θ)

The domain excludes all θ where sin(θ) = 0, i.e., θ ≠ nπ for any integer n. The range is (−∞, −1] ∪ [1, ∞).

Angle unit conversions used internally:

θ_rad = θ_deg × π180

θ_rad = θ_grad × π200

Where θ = angle input value, π ≈ 3.14159265358979. The Pythagorean identity connecting cosecant and cotangent: 1 + cot²(θ) = csc²(θ).

Reference Data

Angle (°)	Angle (rad)	sin(θ)	csc(θ)	Exact Form
0	0	0	Undefined	-
15	π/12	0.258819	3.863703	√6 + √2
30	π/6	0.5	2	2
45	π/4	√22	√2	√2
60	π/3	√32	2√33	2√3/3
90	π/2	1	1	1
120	2π/3	√32	2√33	2√3/3
135	3π/4	√22	√2	√2
150	5π/6	0.5	2	2
180	π	0	Undefined	-
210	7π/6	−0.5	−2	−2
225	5π/4	−√22	−√2	−√2
240	4π/3	−√32	−2√33	−2√3/3
270	3π/2	−1	−1	−1
300	5π/3	−√32	−2√33	−2√3/3
315	7π/4	−√22	−√2	−√2
330	11π/6	−0.5	−2	−2
360	2π	0	Undefined	-

Frequently Asked Questions

At θ = 0°, sin(θ) = 0, and division by zero is mathematically undefined - not infinite. While IEEE 754 floating-point returns Infinity, this calculator applies an epsilon threshold of 10⁻¹² to detect near-zero sine values and correctly reports "Undefined" to prevent misinterpretation in engineering or physics contexts.

JavaScript's Math.sin(π) returns approximately 1.22 × 10⁻¹⁶ rather than exact 0. Without correction, csc(180°) would yield approximately 8.17 × 10¹⁵ - a nonsensical result. The calculator normalizes angles modulo 360°, then checks if the absolute sine value falls below 10⁻¹² to flag it as undefined.

The six trigonometric functions form three reciprocal pairs: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ). The Pythagorean identity 1 + cot²(θ) = csc²(θ) is the cosecant-specific form derived from dividing sin² + cos² = 1 by sin².

Gradians (gon) divide a right angle into 100 equal parts, making a full circle 400grad. This base-10 subdivision is standard in European land surveying and geodesy (ISO 31-1). Percentage slope calculations become trivial: a 1grad angle corresponds to a 1% grade at small angles. The conversion factor is θ_rad = θ_grad × π/200.

Yes. The calculator maintains a lookup table of special angles in the first quadrant (30°, 45°, 60°, 90°) and maps equivalent angles from all four quadrants using reference angle reduction. When a match is found, the result displays both the decimal approximation and the exact form (e.g., 2√3/3 for 60°).

csc(θ) inherits its sign from sin(θ). In Quadrant I (0° - 90°) and Quadrant II (90° - 180°), sine is positive, so cosecant is positive. In Quadrant III (180° - 270°) and Quadrant IV (270° - 360°), sine is negative, so cosecant is negative. The mnemonic "All Students Take Calculus" encodes which functions are positive in each quadrant.