To determine within which range the measure of a central angle, given in degrees, falls when converted to radians, follow these steps:
1. Convert the Angle from Degrees to Radians:
- Use the conversion factor between degrees and radians: [tex]\( 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \)[/tex].
- For an angle of [tex]\(125^\circ\)[/tex]:
[tex]\[
\text{Angle in radians} = 125 \times \frac{\pi}{180}
\][/tex]
- Simplifying this:
[tex]\[
\text{Angle in radians} \approx 2.181661564992912
\][/tex]
2. Determine the Range of the Central Angle in Radians:
Assess which interval the radian measure falls into:
- [tex]\(0 \leq \text{angle in radians} < \frac{\pi}{2}\)[/tex]:
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex]
- Since [tex]\(2.181661\)[/tex] is greater than [tex]\(1.5708\)[/tex], this range is not applicable.
- [tex]\(\frac{\pi}{2} \leq \text{angle in radians} < \pi\)[/tex]:
- [tex]\(\pi \approx 3.1416\)[/tex]
- Since [tex]\(1.5708 \leq 2.181661 < 3.1416\)[/tex], the angle of [tex]\(2.181661\)[/tex] radians falls within this range.
- [tex]\(\pi \leq \text{angle in radians} < \frac{3\pi}{2}\)[/tex]:
- [tex]\(\frac{3\pi}{2} \approx 4.7124\)[/tex]
- [tex]\(2.181661\)[/tex] is less than [tex]\(3.1416\)[/tex], so this range is not applicable.
- [tex]\(\frac{3\pi}{2} \leq \text{angle in radians} < 2\pi\)[/tex]:
- [tex]\(2\pi \approx 6.2832\)[/tex]
- [tex]\(2.181661\)[/tex] is much less than [tex]\(4.7124\)[/tex], hence this range is also not applicable.
Thus, the measure of the central angle [tex]\(2.181661564992912\)[/tex] radians is within the range:
[tex]\[
\boxed{\frac{\pi}{2} \text{ to } \pi \text{ radians}}
\][/tex]
