Question

For what values of $x\in (2\pi ,8\pi ),\sin x\le 0?$

• A. $x\in [3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi ]$
• B. $x\in (3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi )$
• C. $x\in [3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi )$
• D. $x\in (3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi ]$

Answer 1

The correct option is C $x\in [3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi )$

Given, the graph of $y=\sin x:$


$\sin x\le 0\Rightarrow$ The graph of $\sin x$ should lie below the $x-$ axis in the particular interval.
Now, for $x\in (2\pi ,8\pi )$ highlighting the region where $\sin x\le 0:$

From the graph, we can see the values of $x$ for which $\sin x\le 0$ is
$x\in [3\pi ,4\pi ]\cup [5\pi ,6\pi ]\cup [7\pi ,8\pi )$