Certainly! Let's go through the given table and provide a detailed explanation for each [tex]\( x \)[/tex] and its corresponding [tex]\( f(x) \)[/tex] value.
1. When [tex]\( x = 0 \)[/tex]:
- [tex]\( f(x) = 0 \)[/tex]
2. When [tex]\( x = \frac{\pi}{6} \)[/tex]:
- [tex]\( f(x) = 0.86 \)[/tex]
3. When [tex]\( x = \frac{\pi}{3} \)[/tex]:
- [tex]\( f(x) = 0.5 \)[/tex]
4. When [tex]\( x = \frac{\pi}{2} \)[/tex]:
- [tex]\( f(x) = 0 \)[/tex]
5. When [tex]\( x = \frac{2\pi}{3} \)[/tex]:
- [tex]\( f(x) = -0.5 \)[/tex]
6. When [tex]\( x = \frac{5\pi}{6} \)[/tex]:
- [tex]\( f(x) = -0.86 \)[/tex]
7. When [tex]\( x = \pi \)[/tex]:
- [tex]\( f(x) = -1 \)[/tex]
8. When [tex]\( x = \frac{7\pi}{6} \)[/tex]:
- [tex]\( f(x) = -0.86 \)[/tex]
9. When [tex]\( x = \frac{4\pi}{3} \)[/tex]:
- [tex]\( f(x) = -0.5 \)[/tex]
10. When [tex]\( x = \frac{3\pi}{2} \)[/tex]:
- [tex]\( f(x) = 0 \)[/tex]
11. When [tex]\( x = \frac{5\pi}{3} \)[/tex]:
- [tex]\( f(x) = 0.5 \)[/tex]
12. When [tex]\( x = 2\pi \)[/tex]:
- [tex]\( f(x) = 1 \)[/tex]
Let's ensure the values listed in the table are precise. Here, we're translating angle measures in radians to specific function values based on an apparent periodic pattern tied to trigonometric functions:
- The function [tex]\( f(x) \)[/tex] seems to vary smoothly across well-known angles measured in radians.
- Specific values like 0, [tex]\(\pi/2\)[/tex], [tex]\(\pi\)[/tex], etc., provide useful points of reference for typical trigonometric functions, hinting at relationships with sine or cosine waves.
Here's a recapitulated table with the detailed explanations for each point:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 0 \\
\hline
\frac{\pi}{6} & 0.86 \\
\hline
\frac{\pi}{3} & 0.5 \\
\hline
\frac{\pi}{2} & 0 \\
\hline
\frac{2\pi}{3} & -0.5 \\
\hline
\frac{5\pi}{6} & -0.86 \\
\hline
\pi & -1 \\
\hline
\frac{7\pi}{6} & -0.86 \\
\hline
\frac{4\pi}{3} & -0.5 \\
\hline
\frac{3\pi}{2} & 0 \\
\hline
\frac{5\pi}{3} & 0.5 \\
\hline
2\pi & 1 \\
\hline
\end{array}
\][/tex]
This accurately maps the function [tex]\( f(x) \)[/tex] at given [tex]\( x \)[/tex] values in radians, showing how [tex]\( f(x) \)[/tex] oscillates, consistent with periodic behavior commonly seen in trigonometric functions.
