Question

which of the following are in the correct order from least to greatest?
$\frac{3\pi}{10}, 60^\circ, \frac{\pi}{2}, \frac{2\pi}{3}, 255^\circ$
$255^\circ, \frac{2\pi}{3}, \frac{\pi}{2}, 60^\circ, \frac{3\pi}{10}$
$\frac{3\pi}{10}, \frac{\pi}{2}, \frac{2\pi}{3}, 60^\circ, 255^\circ$
$255^\circ, 60^\circ, \frac{2\pi}{3}, \frac{\pi}{2}, \frac{3\pi}{10}$

Explanation:

Step1: Convert all to degrees

To compare, convert radians to degrees using \(1\ \text{radian}=\frac{180^\circ}{\pi}\).

• \(\frac{3\pi}{10}\): \(\frac{3\pi}{10}\times\frac{180^\circ}{\pi}= 54^\circ\)
• \(\frac{\pi}{2}\): \(\frac{\pi}{2}\times\frac{180^\circ}{\pi}= 90^\circ\)
• \(\frac{2\pi}{3}\): \(\frac{2\pi}{3}\times\frac{180^\circ}{\pi}= 120^\circ\)

Step2: List all values in degrees

• Option 1: \(54^\circ, 60^\circ, 90^\circ, 120^\circ, 255^\circ\)
• Option 2: \(255^\circ, 120^\circ, 90^\circ, 60^\circ, 54^\circ\) (reverse)
• Option 3: \(54^\circ, 90^\circ, 120^\circ, 60^\circ, 255^\circ\) (incorrect order)
• Option 4: \(255^\circ, 60^\circ, 120^\circ, 90^\circ, 54^\circ\) (incorrect)

Step3: Check order (least to greatest)

Option 1 has values \(54^\circ < 60^\circ < 90^\circ < 120^\circ < 255^\circ\), which is correct.

Answer:

\(\boldsymbol{\frac{3\pi}{10}, 60^\circ, \frac{\pi}{2}, \frac{2\pi}{3}, 255^\circ}\) (the first option)