To determine the range of the central angle in radians for an arc that measures [tex]\( 125^\circ \)[/tex], let's follow the steps:
1. Convert the arc angle from degrees to radians:
We start by using the conversion formula between degrees and radians:
[tex]\[
\text{radians} = \left(\frac{\pi}{180}\right) \times \text{degrees}
\][/tex]
Substituting [tex]\( 125^\circ \)[/tex] into the formula:
[tex]\[
\text{arc angle in radians} = \left(\frac{\pi}{180}\right) \times 125
\][/tex]
This simplifies to:
[tex]\[
\text{arc angle in radians} = \frac{125\pi}{180}
\][/tex]
Simplifying the fraction further by dividing numerator and denominator by 5:
[tex]\[
\text{arc angle in radians} = \frac{25\pi}{36} \approx 2.1816615649929116
\][/tex]
2. Determine the range of the central angle in radians:
Now, compare the calculated arc angle in radians with the given ranges:
- [tex]\(0 \leq \text{angle} < \frac{\pi}{2}\)[/tex]
- [tex]\(\frac{\pi}{2} \leq \text{angle} < \pi\)[/tex]
- [tex]\(\pi \leq \text{angle} < \frac{3\pi}{2}\)[/tex]
- [tex]\(\frac{3\pi}{2} \leq \text{angle} < 2\pi\)[/tex]
Comparing [tex]\(2.1816615649929116\)[/tex] radians with these ranges:
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex]
- [tex]\(\pi \approx 3.1416\)[/tex]
The value [tex]\( 2.1816615649929116 \)[/tex] radians lies between [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex] and [tex]\(\pi \approx 3.1416\)[/tex].
Thus, the central angle in radians, when the arc measures [tex]\( 125^\circ \)[/tex], falls within the range:
[tex]\[
\boxed{\frac{\pi}{2} \text{ to } \pi \text{ radians}}
\][/tex]
