Question

what is the range of $y = \csc(x)$?
$\bigcirc$ $y \leq -1$
$\bigcirc$ $y \geq 1$
$\bigcirc$ $y \leq -1$ or $y \geq 1$
$\bigcirc$ $-1 \leq y \leq 1$

Explanation:

Step1: Recall the definition of cosecant

The cosecant function is defined as \( \csc(x)=\frac{1}{\sin(x)} \).

Step2: Analyze the range of sine function

The range of \( \sin(x) \) is \( - 1\leqslant\sin(x)\leqslant1 \), and \( \sin(x)
eq0 \) (because \( \csc(x) \) is undefined when \( \sin(x) = 0 \)).

Step3: Analyze the range of reciprocal

Let \( t=\sin(x) \), then \( y = \frac{1}{t} \), where \( - 1\leqslant t\leqslant1 \) and \( t
eq0 \).

• When \( 0\lt t\leqslant1 \), taking reciprocal, we get \( y=\frac{1}{t}\geqslant1 \) (since for a positive fraction between 0 and 1, its reciprocal is greater than or equal to 1).
• When \( - 1\leqslant t\lt0 \), taking reciprocal, we get \( y = \frac{1}{t}\leqslant - 1 \) (since for a negative fraction between - 1 and 0, its reciprocal is less than or equal to - 1).

So the range of \( y = \csc(x) \) is \( y\leqslant - 1 \) or \( y\geqslant1 \).

Answer:

\( y\leqslant - 1 \) or \( y\geqslant1 \) (the option with the text " \( y\leqslant - 1 \) or \( y\geqslant1 \)")