Cosecant Calculator
Calculate csc θ = 1 / sin θ for any angle in degrees, radians or multiples of π, with exact surd values for standard angles.
The cosecant of an angle is the reciprocal of its sine: csc θ = 1 / sin θ. In a right triangle it is the hypotenuse divided by the side opposite the angle. Enter an angle in degrees, radians or multiples of π to get csc θ to ten significant digits, the exact surd form for standard angles, and the sine it was calculated from.
How to use this tool
- Type the angle.
- Choose its unit: degrees, radians, or a multiple of π (enter 0.25 for π/4).
- Read csc θ, its exact form when the angle is a standard one, and sin θ.
Formula
csc θ = 1 / sin θ = hypotenuse / opposite
Cosecant is undefined wherever sin θ = 0, which is at 0°, 180°, 360° and every other multiple of 180° (kπ radians). Its values never lie strictly between −1 and 1: |csc θ| ≥ 1 for every angle where it exists, because |sin θ| ≤ 1.
Example
For θ = 30°, sin 30° = 0.5, so csc 30° = 1 / 0.5 = 2. For θ = 1 radian, sin 1 ≈ 0.8414709848, so csc 1 ≈ 1.188395106. At 45° the exact value is √2 ≈ 1.414213562, and at 270° (1.5π) it is −1.
Frequently asked questions
Is cosecant the same as inverse sine?
No. csc θ = 1 / sin θ is a reciprocal. Inverse sine (arcsin, written sin⁻¹) goes the other way: it takes a ratio and returns an angle. The ⁻¹ on a calculator key means inverse function, not 1 divided by sine.
How do I calculate csc on a calculator without a csc button?
Work out sin θ, then press 1/x or divide 1 by the result. Make sure the calculator is in the same angle mode, degrees or radians, as your angle.
Why is csc 180° undefined?
Because sin 180° = 0 and dividing by zero has no value. The graph of y = csc x has a vertical asymptote at every multiple of 180°.
What are the common cosecant values?
csc 30° = 2, csc 45° = √2 ≈ 1.4142, csc 60° = 2√3/3 ≈ 1.1547 and csc 90° = 1. From 90° to 180° they repeat in reverse (csc 150° = csc 30°), and from 180° to 360° they are negative.
Last updated: 2026-10-30