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 \(1\ \text{radian}=\frac{180^\circ}{\pi}\).

• \(\frac{3\pi}{10}\): \(\frac{3\pi}{10}\times\frac{180^\circ}{\pi}= 54^\circ\)
• \(\frac{\pi}{2}\): \(\frac{\pi}{2}\times\frac{180^\circ}{\pi}= 90^\circ\)
• \(\frac{2\pi}{3}\): \(\frac{2\pi}{3}\times\frac{180^\circ}{\pi}= 120^\circ\)
• \(60^\circ\) remains \(60^\circ\), \(255^\circ\) remains \(255^\circ\).

Step2: Order the degrees

Now list all values in degrees: \(54^\circ\) (\(\frac{3\pi}{10}\)), \(60^\circ\), \(90^\circ\) (\(\frac{\pi}{2}\)), \(120^\circ\) (\(\frac{2\pi}{3}\)), \(255^\circ\).
Order from least to greatest: \(54^\circ\) (\(\frac{3\pi}{10}\)), \(60^\circ\), \(90^\circ\) (\(\frac{\pi}{2}\)), \(120^\circ\) (\(\frac{2\pi}{3}\)), \(255^\circ\).
Check the options: The first option is \(\frac{3\pi}{10}, 60^\circ, \frac{\pi}{2}, \frac{2\pi}{3}, 255^\circ\), which matches the ordered degrees.

Answer:

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