To determine the correct order from greatest to least among the given angles, we need to convert them all to the same unit. In this case, radians can be a convenient choice because all angles except for [tex]\(330^\circ\)[/tex] are already given in radians.
First, let's convert [tex]\(330^\circ\)[/tex] to radians:
[tex]\[ 330^\circ = 330 \times \frac{\pi}{180} = \frac{11\pi}{6} \][/tex]
Now, let’s list all angles in radians:
- [tex]\(\frac{\pi}{2}\)[/tex]
- [tex]\(\frac{2\pi}{3}\)[/tex]
- [tex]\(\frac{7\pi}{6}\)[/tex]
- [tex]\(\frac{5\pi}{3}\)[/tex]
- [tex]\(\frac{11\pi}{6}\)[/tex]
To compare these angles, we can sort their coefficients:
1. [tex]\(\frac{5\pi}{3}\)[/tex] - coefficient is [tex]\(\frac{5}{3} \approx 1.67\)[/tex]
2. [tex]\(\frac{11\pi}{6}\)[/tex] - coefficient is [tex]\(\frac{11}{6} \approx 1.83\)[/tex]
3. [tex]\(\frac{7\pi}{6}\)[/tex] - coefficient is [tex]\(\frac{7}{6} \approx 1.17\)[/tex]
4. [tex]\(\frac{2\pi}{3}\)[/tex] - coefficient is [tex]\(\frac{2}{3} \approx 0.67\)[/tex]
5. [tex]\(\frac{\pi}{2}\)[/tex] - coefficient is [tex]\(\frac{1}{2} = 0.5\)[/tex]
Thus sorted from greatest to least in coefficients, the correct order from greatest to least is:
[tex]\[ \frac{5\pi}{3}, \frac{11\pi}{6}, \frac{7\pi}{6}, \frac{2\pi}{3}, \frac{\pi}{2} \][/tex]
If we look at the provided options:
- [tex]\(\frac{5\pi}{3}\)[/tex]
- [tex]\(\frac{7\pi}{6}\)[/tex]
- [tex]\(\frac{2\pi}{3}\)[/tex]
- [tex]\(\frac{\pi}{2}\)[/tex]
- [tex]\(330^\circ = \frac{11\pi}{6}\)[/tex]
The correct order from greatest to least in the options given is:
[tex]\( \boxed{330^\circ, \frac{5\pi}{3}, \frac{7\pi}{6}, \frac{2\pi}{3}, \frac{\pi}{2}} \)[/tex]
