Cosecant Calculator

The cosecant is the reciprocal of sine:

csc(θ) = 1 / sin(θ)

It exists at every angle except multiples of 180° (0°, 180°, 360°, …), where sin(θ) = 0 and the function is undefined. At those angles the calculator shows “undefined.”

Key values to know:

• csc(30°) = 2 (since sin 30° = 0.5)
• csc(45°) = √2 ≈ 1.414
• csc(60°) = 2√3/3 ≈ 1.155
• csc(90°) = 1 (the minimum absolute value)

Range: cosecant never falls between −1 and +1. If you compute csc and get a number between −1 and 1 exclusive, something went wrong. The function lives in (−∞, −1] ∪ [1, +∞).

Period: 360° (same as sine). The graph has vertical asymptotes at every multiple of 180° and U-shaped curves opening up in the first/second quadrants and down in the third/fourth.

The “co-” prefix comes from “complementary.” Cosecant of θ equals the secant of the complement: csc(θ) = sec(90° − θ). This is the same relationship that links sine and cosine.

Cosecant appears in integration. The classic result is ∫csc(θ) dθ = ln|tan(θ/2)| + C, which nobody guesses on sight: it falls out of the half-angle substitution and looks like an accident until you see the derivation. It also turns up in optics (the cosecant-squared antenna radiation pattern), in ballistics, and in surveying formulas where angles are measured from vertical rather than horizontal.

The calculator also shows the conversion between degrees and radians for the entered angle.