To compare an angle having a measure of [tex]\( 120^\circ \)[/tex] with an angle measuring [tex]\( \frac{5\pi}{6} \)[/tex] radians, we need to express both angles in the same unit, either in degrees or in radians. For this comparison, we will convert the angle of [tex]\( 120^\circ \)[/tex] into radians.
1. Convert [tex]\( 120^\circ \)[/tex] to radians:
We use the conversion factor [tex]\( \pi \)[/tex] radians = [tex]\( 180^\circ \)[/tex].
[tex]\[
\text{Angle in radians} = 120^\circ \times \left( \frac{\pi \text{ radians}}{180^\circ} \right)
\][/tex]
Simplifying the fraction:
[tex]\[
\text{Angle in radians} = 120 \times \left( \frac{\pi}{180} \right)
\][/tex]
[tex]\[
\text{Angle in radians} = \frac{120\pi}{180}
\][/tex]
[tex]\[
\text{Angle in radians} = \frac{2\pi}{3}
\][/tex]
Thus, the angle of [tex]\( 120^\circ \)[/tex] is equivalent to [tex]\( \frac{2\pi}{3} \)[/tex] radians.
2. Compare [tex]\( \frac{2\pi}{3} \)[/tex] radians with [tex]\( \frac{5\pi}{6} \)[/tex] radians:
We need to see if these two measures are equal. To do this more precisely, we compare their numerical values.
Calculate the numerical value of each angle in radians using the value of [tex]\( \pi \approx 3.14159 \)[/tex]:
- For [tex]\( \frac{2\pi}{3} \)[/tex]:
[tex]\[
\frac{2 \times 3.14159}{3} \approx 2.0943951023931953
\][/tex]
- For [tex]\( \frac{5\pi}{6} \)[/tex]:
[tex]\[
\frac{5 \times 3.14159}{6} \approx 2.6179938779914944
\][/tex]
3. Compare the numerical values:
- The value of [tex]\( \frac{2\pi}{3} \)[/tex] radians is approximately [tex]\( 2.0943951023931953 \)[/tex].
- The value of [tex]\( \frac{5\pi}{6} \)[/tex] radians is approximately [tex]\( 2.6179938779914944 \)[/tex].
From these calculations, we can see that [tex]\( 2.0943951023931953 \)[/tex] is not equal to [tex]\( 2.6179938779914944 \)[/tex]. Therefore, the angle measuring [tex]\( 120^\circ \)[/tex] (or [tex]\( \frac{2\pi}{3} \)[/tex] radians) is not the same as the angle measuring [tex]\( \frac{5\pi}{6} \)[/tex] radians.
Hence, an angle of [tex]\( 120^\circ \)[/tex] is different from an angle of [tex]\( \frac{5\pi}{6} \)[/tex] radians.
