Cosecant Calculator
Calculate the cosecant of an angle in degrees, radians or gradians. The cosecant is the reciprocal of the sine.
What is Cosecant?
Cosecant is the reciprocal of sine: csc(θ) = 1 / sin(θ). Looking at a right triangle, it gives the ratio of the hypotenuse to the opposite side.
Among the notable values are csc(30°) = 2, csc(45°) ≈ 1.4142 and csc(90°) = 1. Wherever sin(θ) = 0 the cosecant becomes undefined, namely at 0°, 180°, 360° and any integer multiple of 180°.
Unit conversions: 1° = π/180 rad ≈ 0.01745 rad. Going the other way, 1 rad = 180°/π ≈ 57.296°. And 400 gradians equal 360°.
The cosecant function
Cosecant is just the sine flipped over: csc(θ) = 1/sen(θ). It repeats every 2π, and wherever the sine drops to zero you get a vertical asymptote, which happens at θ = kπ for any integer k. The function is odd, so csc(−θ) = −csc(θ). A few values worth memorizing: csc(30°) = 2, csc(45°) = √2, csc(60°) = 2√3/3 and csc(90°) = 1. The identity you'll lean on most is 1 + cot²θ = csc²θ, and its derivative is (csc x)' = −csc x · cot x.
Applications: calculus, optics and engineering
You run into the cosecant more often than you'd expect. It shows up in integral calculus through the antiderivative ∫ csc x dx = −ln|csc x + cot x| + C, in geometric optics when you measure ray angles against a surface, and in Maclaurin series for trigonometric expansions. Mechanical design uses it too: gear tooth profiles, cam followers, and ramps all work out the travel distance from a height through the factor 1/sen(θ). Accessibility standards like Brazil's NBR 9050 rely on that same relation to figure ramp length from a target rise.
FAQ
Why is csc(0) undefined? Since sen(0) = 0, you'd be dividing by zero, which has no value. Every θ = kπ hits the same wall.
What is the minimum value where csc is positive? That floor is csc(90°) = 1. Move away from there and the function climbs without bound as θ heads toward a multiple of π.
How does csc relate to cot? The link is the Pythagorean identity 1 + cot²θ = csc²θ. You get it by taking sen²θ + cos²θ = 1 and dividing through by sen²θ.
Is csc odd or even? It's odd, csc(−θ) = −csc(θ), a trait it picks up straight from the sine.
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Calculate the cosecant of an angle
The cosecant is simply 1 divided by the angle's sine, that is, the inverse of the sine. It rounds out the group of reciprocal functions alongside the secant and the cotangent, and tends to show up in calculus and in more advanced problems. Here you get that value for any angle.
Pick the angle's unit first (degrees, radians or gradians). The tool computes the cosecant and handles the points where the sine hits zero, cases in which the function no longer exists. Since an ordinary calculator almost never has a button for this, you skip the chore of finding the sine and inverting it by hand.
Everything is computed in your own browser. It works as a quick reference for trigonometry, calculus and engineering.