About

Reciprocal trigonometric functions - csc, sec, and cot - are critical in calculus, signal processing, and structural load analysis, yet they introduce discontinuities that standard calculators handle poorly. A computation at θ = 90° yields sec(θ) → undefined, and mistaking a near-infinite floating-point result for a valid output can propagate catastrophic error through an entire engineering model. This calculator detects those singularities explicitly using an epsilon threshold of 1×10⁻¹⁰ and reports undefined rather than a misleading large number.

Angles are accepted in degrees, radians, or gradians and internally normalized before computation. The tool assumes a pure mathematical context: no atmospheric refraction, no relativistic corrections. For angles entered in degrees, conversion precision is limited by IEEE 754 double-precision representation of π, introducing a maximum relative error on the order of 10⁻¹⁵. Pro tip: when working near known singularities (multiples of π or 180°), verify your input unit carefully. A 1° mistake near 0° swings csc from undefined to 57.3.

Formulas

The three reciprocal trigonometric functions are defined as follows:

csc(θ) = 1sin(θ) , sec(θ) = 1cos(θ) , cot(θ) = cos(θ)sin(θ)

Where θ is the input angle. Angle unit conversions applied before evaluation:

θ_rad = θ_deg × π180 , θ_rad = θ_grad × π200

Singularity detection: if |sin(θ)| < 10⁻¹⁰, then csc and cot are reported as undefined. Likewise, if |cos(θ)| < 10⁻¹⁰, then sec is undefined.

Pythagorean identities linking these functions:

1 + cot²(θ) = csc²(θ) , 1 + tan²(θ) = sec²(θ)

Reference Data

Angle (°)	Angle (rad)	sin	cos	csc	sec	cot
0	0	0	1	undef	1	undef
15	π/12	0.2588	0.9659	3.8637	1.0353	3.7321
30	π/6	0.5	0.8660	2	1.1547	1.7321
45	π/4	0.7071	0.7071	1.4142	1.4142	1
60	π/3	0.8660	0.5	1.1547	2	0.5774
75	5π/12	0.9659	0.2588	1.0353	3.8637	0.2679
90	π/2	1	0	1	undef	0
120	2π/3	0.8660	−0.5	1.1547	−2	−0.5774
135	3π/4	0.7071	−0.7071	1.4142	−1.4142	−1
150	5π/6	0.5	−0.8660	2	−1.1547	−1.7321
180	π	0	−1	undef	−1	undef
210	7π/6	−0.5	−0.8660	−2	−1.1547	1.7321
225	5π/4	−0.7071	−0.7071	−1.4142	−1.4142	1
240	4π/3	−0.8660	−0.5	−1.1547	−2	0.5774
270	3π/2	−1	0	−1	undef	0
300	5π/3	−0.8660	0.5	−1.1547	2	−0.5774
315	7π/4	−0.7071	0.7071	−1.4142	1.4142	−1
330	11π/6	−0.5	0.8660	−2	1.1547	−1.7321
360	2π	0	1	undef	1	undef

Frequently Asked Questions

At exact singularities (e.g., csc at 0° or 180°), the denominator sin(θ) equals zero and the function has a vertical asymptote - no finite value exists. Returning a large float like ±1×10¹⁵ would be mathematically incorrect and could silently corrupt downstream calculations. This calculator uses an epsilon threshold of 1×10⁻¹⁰ to detect near-zero denominators and explicitly reports "undefined" to prevent such errors.

IEEE 754 double-precision stores π as an approximation (~3.141592653589793), so sin(180°) computes as approximately 1.2×10⁻¹⁶ rather than exact zero. Without the epsilon guard, the calculator would return csc(180°) ≈ 8.2×10¹⁵ - a nonsensical result. The 10⁻¹⁰ threshold is deliberately conservative: tight enough to preserve 10-digit accuracy for legitimate inputs, but wide enough to catch IEEE rounding artifacts at known poles.

One full revolution equals 360° or 400 gradians (grad). Gradians divide a right angle into exactly 100 units, making them natural for land surveying and civil engineering where percentage-grade slopes map directly to gradian measures. For example, a 1% slope equals 1 grad. If your source data comes from a theodolite or European surveying instrument, use gradians.

No. Since csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = cos(θ)/sin(θ), the first two are reciprocals of bounded functions with range [−1, 1], so their outputs always satisfy |value| ≥ 1 (or are undefined). Cotangent crosses zero whenever cos(θ) = 0, which occurs at θ = 90° + n·180°. So cot can be zero, but csc and sec cannot.

Use the Pythagorean identity: 1 + cot²(θ) must equal csc²(θ) for any angle where both are defined. Similarly, 1 + tan²(θ) = sec²(θ). Compute both sides and check that the difference is within machine epsilon (~10⁻¹⁴). The results panel in this tool displays sin, cos, and tan alongside the reciprocal values specifically to facilitate this cross-check.

At 90°, sin(90°) = 1 (nonzero), so cot(90°) = cos(90°)/sin(90°) = 0/1 = 0. Cotangent is undefined only when sin(θ) = 0, which occurs at 0°, 180°, 360°, etc. The fact that tan(90°) is undefined (cos = 0) does not make cot(90°) undefined - they have different singularity sets.