Sketch the graph of \(y = \csc(x)\), \(0 \leq x \leq 2\pi\)

1. Since \(\csc(x) = \dfrac{1}{\sin(x)}\), sketch \(y = \sin(x)\), \(0 \leq x \leq 2\pi\).

2. Identify the points which are on the graphs of both \(y = \sin(x)\) and \(y = \csc(x)\). We want the values of \(x\) for which \(\sin(x) = 1\) or \(\sin(x) = -1\). For \(0 \lt x \leq 2\pi\), \(\sin(\frac{\pi}{2}) = 1\) and \(\sin(\frac{3\pi}{2}) = -1\). The points \(\left( \dfrac{\pi}{2} , 1\right)\) and \(\left( \dfrac{3\pi}{2}, -1 \right)\) are on both functions. Plot these points.

3. Identify the \(x\)-intercepts of \(y = \sin(x)\). At these values of \(x\), \(y = \csc(x)\) is undefined and there will be vertical asymptotes at \(x=0\), \(x = \pi\), and \(x = 2\pi\).

Sketch the graph of \(y = \csc(x)\), \(0 \leq x \leq 2\pi\)

4. Determine what happens to the reciprocal function between the asymptotes.

• As the value of \(x\) moves from \(\dfrac{\pi}{2}\) towards \(0\), the value of \(\sin(x)\) goes from \(1\) to \(0\). Then the value of the reciprocal goes from \(1\) to \(\infty\).
• As the value of \(x\) moves from \(\dfrac{\pi}{2}\) towards \(1\), the value of \(\sin(x)\) goes from \(1\) to \(0\). Then the value of the reciprocal goes from \(1\) to \(\infty\).
• As the value of \(x\) moves from \(\dfrac{3\pi}{2}\) towards \(\pi\), the value of \(\sin(x)\) goes from \(-1\) to \(0\), approaching from below. Then the value of the reciprocal goes from \(-1\) to \(- \infty\).

• As the value of \(x\) moves from \(\dfrac{3\pi}{2}\) towards \(2\pi\), the value of \(\sin(x)\) goes from \(-1\) to \(0\), approaching from below. Then the value of the reciprocal goes from \(-1\) to \(- \infty\).

Properties of \(y = \csc(x)\)

It is straightforward to see that the cosecant function will be periodic and have the same period as the sine function. For the cosecant function, state the \(x\) and \(y\)-intercepts, the period and the equations of any asymptotes, and the domain and range.

There are no \(x\)-intercepts.

There is no \(y\)-intercept.

The period of \(y = \csc(x)\) is \(2\pi\).

The vertical asymptotes occur every \(\pi\) units with one of them at \(x = 0\). Therefore, the equations of the vertical asymptotes are \(x = n \pi, n \in \mathbb{Z}\). There are no horizontal asymptotes. The domain is \(\left\{ x \mid x \neq n\pi , x \in \mathbb{R} , n \in \mathbb{Z} \right\}\).

No values of \(y = \csc(x)\) lie between \(-1\) and \(1\). The range of \(y = \csc(x)\) is \(\left\{ y \mid y \leq -1 \text{ or } y \geq 1, y \in \mathbb{R} \right\}\).

The range can also be written using absolute value notation as \(\left\{ y \mid \lvert y \rvert \geq 1 , y \in \mathbb{R} \right\}\).