Free Cosecant Calculator

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Cosecant is one of the three reciprocal trigonometric functions, defined as csc(θ) = 1/sin(θ). It appears throughout physics and engineering wherever a relationship is naturally expressed in terms of "1 over sine" — for instance, in analyzing the mechanical advantage of an inclined plane, or in astronomy when converting between an object's altitude angle and the path length of light or radio signal passing through the atmosphere (air mass roughly follows a cosecant law near the horizon). Because cosecant is undefined wherever sine is zero — at 0°, 180°, 360°, and every 180° interval from there — this calculator flags those cases explicitly rather than returning a misleadingly huge number.

How it works

Enter an angle and choose degrees or radians. The calculator first evaluates sin(θ), then takes its reciprocal, 1/sin(θ), returning "undefined" instead of an error whenever sin(θ) is zero. Like sine, cosecant is an odd function, so csc(-θ) = -csc(θ).

Enter angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

csc(30°)

sin(30°) = 0.5, so csc(30°) = 1/0.5 = 2.

csc(210°)

sin(210°) = -0.5, so csc(210°) = 1/(-0.5) = -2, negative because 210° falls in the third quadrant where sine is negative.

csc(180°)

csc(180°) is undefined, because sin(180°) = 0.

Common mistakes to avoid

Entering an angle of 0°, 180°, or 360° and expecting a numeric result instead of "undefined".
Confusing cosecant with secant — cosecant is 1/sine, secant is 1/cosine.
Assuming cosecant's range includes values between -1 and 1 — it never does.
Mixing up csc⁻¹(x) (an inverse trig function returning an angle) with 1/csc(x) (which just equals sin(x)).

Frequently asked questions

Cosecant is simply the reciprocal of sine: csc(θ) = 1/sin(θ). As sine approaches 0 near 0° or 180°, cosecant grows without bound in either the positive or negative direction.

Only values ≤ -1 or ≥ 1 — since sine is always between -1 and 1, its reciprocal can never land strictly between -1 and 1.

No — that's a frequent confusion. Arcsin answers "which angle produces this sine value?" and returns an angle. Cosecant takes an angle and returns 1 divided by its sine.

Via the Pythagorean identity 1 + cot²(θ) = csc²(θ), the counterpart to the more familiar 1 + tan²(θ) = sec²(θ).

No — since csc(θ) = 1/sin(θ), that would require sin(θ) to be infinite, which never happens for a real angle.

Whenever a quantity is inversely proportional to how "steep" an angle is (like force components on an incline, or signal path length through a layer of atmosphere), cosecant often gives a cleaner formula than repeatedly writing 1/sin(θ).

Cosecant's graph has a repeating series of U-shaped and n-shaped branches with vertical asymptotes everywhere sine crosses zero — essentially the "outside" curves that hug sine's wave from above and below.

Odd — csc(-θ) = -csc(θ), which mirrors the same property in sine, since cosecant is defined directly as sine's reciprocal.