To determine within which range the measure of the central angle of an arc measuring [tex]\( 125^{\circ} \)[/tex] falls, follow these steps:
1. Convert Degrees to Radians:
The conversion formula from degrees to radians is given by
[tex]\[
\text{radians} = \text{degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Plugging in [tex]\( 125^\circ \)[/tex]:
[tex]\[
\text{radians} = 125^\circ \times \left( \frac{\pi}{180} \right)
\][/tex]
Calculating this gives us approximately [tex]\( 2.181661564992912 \)[/tex] radians.
2. Identify the Radian Range:
We now need to determine which range this angle falls into among the given options:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3 \pi}{2}\)[/tex] radians
- [tex]\(\frac{3 \pi}{2}\)[/tex] to [tex]\(2 \pi\)[/tex] radians
Approximate values for these ranges in decimal form:
- [tex]\(\frac{\pi}{2}\)[/tex] is about [tex]\( 1.5708 \)[/tex]
- [tex]\(\pi\)[/tex] is about [tex]\( 3.1416 \)[/tex]
- [tex]\(\frac{3 \pi}{2}\)[/tex] is about [tex]\( 4.7124 \)[/tex]
- [tex]\(2 \pi\)[/tex] is about [tex]\( 6.2832 \)[/tex]
Seeing that [tex]\( 2.181661564992912 \)[/tex] radians falls between [tex]\( \frac{\pi}{2} \)[/tex] and [tex]\(\pi\)[/tex]:
[tex]\[
1.5708 < 2.181661564992912 \leq 3.1416
\][/tex]
3. Conclusion:
Therefore, the measure of the central angle in radians is within the range
[tex]\[
\frac{\pi}{2} \text{ to } \pi \text{ radians}.
\][/tex]
