Cosecant Graph: y = csc x

The cosecant function, written \(y = \csc x\), is the reciprocal of the sine function: \(\csc x = \frac{1}{\sin x}\). Because cosecant is built directly from sine, its graph is closely tied to the sine curve, but it behaves very differently near the places where sine equals zero.

Unlike sine and cosine, which are smooth waves that touch every value between negative 1 and 1, the cosecant graph is made of separate U-shaped branches that never cross the x-axis and shoot off toward infinity near certain angles.

Since \(\csc x = \frac{1}{\sin x}\), the function is undefined wherever \(\sin x = 0\), which happens at every multiple of \(\pi\): \(x = 0, \pm\pi, \pm2\pi, \dots\). At each of these values the cosecant graph has a vertical asymptote.

• Domain: all real numbers except \(x = n\pi\), where \(n\) is any integer.
• Range: \(y \le -1\) or \(y \ge 1\) (cosecant never takes values strictly between negative 1 and 1).
• Period: \(2\pi\), the same period as sine.
• Vertical asymptotes: at \(x = n\pi\) for every integer \(n\).

If you need a refresher on working with angles in radians before locating these asymptotes, see converting between degrees and radians.

1. Sketch the sine curve \(y = \sin x\) lightly as a guide, since the cosecant branches sit on top of it.
2. Mark a vertical asymptote at every x-intercept of sine: \(x = 0, \pm\pi, \pm2\pi, \dots\).
3. Wherever sine reaches a maximum of 1, cosecant touches a local minimum of 1 at the same x-value.
4. Wherever sine reaches a minimum of negative 1, cosecant touches a local maximum of negative 1 at the same x-value.
5. Between each pair of neighboring asymptotes, draw a U-shaped branch opening away from the x-axis, passing through that local minimum or maximum point.

This close relationship between the sine graph and the cosecant graph is the same idea used to build the secant graph from cosine, so once you understand one reciprocal graph the other follows the same pattern.

Graph \(y = 2\csc\left(x - \frac{\pi}{4}\right)\).

Compare this to the parent function \(y = \csc x\): the 2 out front stretches every branch vertically, and the \(\frac{\pi}{4}\) shifts the whole graph to the right by \(\frac{\pi}{4}\).

• New asymptotes occur where \(x - \frac{\pi}{4} = n\pi\), so \(x = \frac{\pi}{4} + n\pi\).
• The local minimum and maximum values move from \(\pm 1\) to \(\pm 2\), since every output is multiplied by 2.
• The shape of each branch is unchanged, just shifted and stretched.

For more general rules on shifting, stretching, and reflecting trig graphs like this one, see transformations of trigonometric functions.

Sometimes you are given a cosecant graph and asked to find its equation instead of the other way around. In that case, look at the asymptotes to find the period and horizontal shift, and look at the local minimum or maximum values to find the vertical stretch, the same reasoning used when working out equations from other trig graphs.

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