Graph of cosecant: y = csc x

Q: Where are the asymptotes of the cosecant graph?

At every multiple of π, where sin x = 0.

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The cosecant graph is 1/sin x: alternating upward and downward U-shapes that touch y = 1 and y = −1 at the peaks and troughs of the sine wave, with vertical asymptotes at every multiple of π.

Open in calculator

The graph of

y=csc(x)

for −7 ≤ x ≤ 7. Dots mark its turning points.

Key features

Domain: All real x except x = nπ
Range: y ≤ −1 or y ≥ 1
Period: 2π
Vertical asymptotes: x = nπ
Local minima: y = 1 at x = π/2 + 2nπ
Local maxima: y = −1 at x = 3π/2 + 2nπ
Symmetry: Odd: csc(−x) = −csc x
y-intercept: None: x = 0 is an asymptote
Derivative: −csc(x)cot(x)
Antiderivative: −ln(|csc(x)+cot(x)|)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of csc x looks like

Cosecant is to sine what secant is to cosine. Each upward U sits on a peak of sin x and touches y = 1 there; each downward U sits on a trough and touches y = −1. Wherever sin x = 0, at every multiple of π, there is an asymptote.

Because sin x is odd, so is csc x: the graph is symmetric about the origin, with an upward branch just right of zero and a downward branch just left of it.

The secant graph shifted right by π/2 is exactly the cosecant graph, since csc x = sec(x − π/2).

Points of interest

Found numerically to about twelve digits in the window −7 ≤ x ≤ 7, then shown in exact form where one exists.

x-intercepts (roots): None in this window
Local maxima: (−1.5708, −1), (4.7124, −1)
Local minima: (−4.7124, 1), (1.5708, 1)
Inflection points: None in this window

There are no roots: csc x is never zero.

Table of values

A few points to plot, with exact values where they are simple.

Values of csc x
x	y = csc x
π/6	2
π/4	√2
π/3	2√3/3
π/2	1
2π/3	2√3/3
5π/6	2
3π/2	−1

How to graph csc x by hand

1
Lightly sketch y = sin x.
2
Draw dashed asymptotes at every zero of sin x: x = 0, ±π, ±2π.
3
Draw an upward U on each peak of the sine curve and a downward U on each trough.
4
Make every branch approach its two asymptotes.