To solve when [tex]\( \sin(3\varnothing) \leq 0 \)[/tex] within a given interval, we start by focusing on the behavior and key points of the sine function.
The sine function, [tex]\(\sin(\theta)\)[/tex], has a period of [tex]\(2\pi\)[/tex]. Given [tex]\(3\varnothing\)[/tex], we know the period changes accordingly to [tex]\(\frac{2\pi}{3}\)[/tex], due to the frequency modification. This means the full cycle of [tex]\(\sin(3\varnothing)\)[/tex] repeats every [tex]\(\frac{2\pi}{3}\)[/tex].
Here's the step-by-step process to address the key points for [tex]\(\sin(3\varnothing) \leq 0\)[/tex] and where we should mark intervals:
1. Understanding Key Points:
[tex]\(\sin(\theta) = 0\)[/tex] at multiples of [tex]\(\pi\)[/tex]: at [tex]\(\theta = 0, \pi, 2\pi, ...\)[/tex].
Therefore, [tex]\( \sin(3\varnothing) = 0\)[/tex] when:
[tex]\(
3\varnothing = 0, \pi, 2\pi, \dots
\)[/tex]
Solving for [tex]\(\varnothing\)[/tex], we get:
[tex]\(
\varnothing = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \dots
\)[/tex]
2. Determining Inequality [tex]\(\leq 0\)[/tex]:
The sine function is negative within intervals where it falls below the x-axis. We are interested in knowing when [tex]\( \sin(3\varnothing) \leq 0\)[/tex]:
Let's examine a period of [tex]\(\sin(3\varnothing)\)[/tex] from [tex]\(\varnothing = 0\)[/tex] to [tex]\(\varnothing = \frac{2\pi}{3}\)[/tex].
During this interval, the sine function goes from [tex]\(0\)[/tex] to [tex]\(1\)[/tex] to [tex]\(0\)[/tex] to [tex]\(-1\)[/tex] to [tex]\(0\)[/tex]. The particular intervals where [tex]\(\sin(3\varnothing) \leq 0\)[/tex] are:
[tex]\[
\left[ \frac{\pi}{3} + 2k\left(\frac{\pi}{3}\right), \frac{2\pi}{3} + 2k\left(\frac{\pi}{3}\right) \right]
\][/tex]
where [tex]\(k\)[/tex] is an integer.
Translating this into the smallest range, we acquire:
[tex]\[
\sin(3\varnothing) \leq 0 \quad \text{for} \quad \varnothing \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right)
\][/tex]
3. Intervals for Solution:
Therefore, periodic intervals where [tex]\(\sin(3\varnothing) \leq 0\)[/tex] within our interval can be expressed as:
[tex]\[
\left[ \frac{\pi}{3} + \frac{2k\pi}{3}, \frac{2\pi}{3} + \frac{2k\pi}{3} \right)
\][/tex]
for integer values of [tex]\(k\)[/tex].
Given this information, it highlights the necessary intervals for when considering [tex]\(\sin(3\varnothing) \leq 0\)[/tex] for graphing purposes.
Hence, the multiples of [tex]\(\pi\)[/tex] and specifically expressed intervals can help identify these concepts for graphing:
- Key multiples: [tex]\(\frac{\pi}{3}\)[/tex]
- Intervals: [tex]\(\left[\frac{\pi}{3}, \frac{2\pi}{3}\right)\)[/tex]
