Question

which of the following are in correct order from greatest to least?
○ $\frac{\pi}{2}$, $330^\circ$, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, $\frac{2\pi}{3}$
○ $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, $\frac{2\pi}{3}$, $\frac{\pi}{2}$, $330^\circ$
○ $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\frac{7\pi}{6}$, $\frac{5\pi}{3}$, $330^\circ$
○ $330^\circ$, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, $\frac{2\pi}{3}$, $\frac{\pi}{2}$

Explanation:

Step1: Convert all to degrees

To compare, convert radians to degrees using \( \theta_{degrees} = \theta_{radians} \times \frac{180^\circ}{\pi} \).

• \( \frac{\pi}{2} \times \frac{180^\circ}{\pi} = 90^\circ \)
• \( \frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 300^\circ \)
• \( \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ \)
• \( \frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ \)
• \( 330^\circ \) remains \( 330^\circ \).

Step2: List all degrees

Now we have: \( 90^\circ, 330^\circ, 300^\circ, 210^\circ, 120^\circ \) (from first option), or for the fourth option: \( 330^\circ, 300^\circ, 210^\circ, 120^\circ, 90^\circ \).

Step3: Sort from greatest to least

Greatest to least: \( 330^\circ > 300^\circ (\frac{5\pi}{3}) > 210^\circ (\frac{7\pi}{6}) > 120^\circ (\frac{2\pi}{3}) > 90^\circ (\frac{\pi}{2}) \). This matches the fourth option.

Answer:

330°, $\frac{5\pi}{3}$, $\frac{7\pi}{6}$, $\frac{2\pi}{3}$, $\frac{\pi}{2}$