Question

compare an angle having a measure of 120° with that of an angle whose measure is \\(\frac{5\pi}{6}\\) radians. explain your reasoning.

Explanation:

Step1: Convert 120° to radians

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). So for 120°, we have \( 120 \times \frac{\pi}{180} = \frac{2\pi}{3} \) radians.

Step2: Convert \( \frac{5\pi}{6} \) to degrees (or compare the two radians)

To convert radians to degrees, we use \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \). For \( \frac{5\pi}{6} \), that's \( \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ \). Now we can compare the two angles. \( \frac{2\pi}{3} \) (which is 120°) is less than \( \frac{5\pi}{6} \) (which is 150°) because \( \frac{2\pi}{3}=\frac{4\pi}{6} \) and \( \frac{4\pi}{6}<\frac{5\pi}{6} \), or in degrees, 120° < 150°.

Answer:

The angle of \( 120^\circ \) (which is \( \frac{2\pi}{3} \) radians) is smaller than the angle of \( \frac{5\pi}{6} \) radians (which is \( 150^\circ \)). This is because when converting \( 120^\circ \) to radians we get \( \frac{2\pi}{3}=\frac{4\pi}{6} \), and \( \frac{4\pi}{6}<\frac{5\pi}{6} \) (or in degrees, \( 120^\circ < 150^\circ \)).