To determine the range within which the measure of the central angle in radians falls, we'll progress through the following steps:
1. Convert the angle from degrees to radians:
The formula to convert degrees to radians is:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
Given that the angle is [tex]\(125^\circ\)[/tex]:
[tex]\[
125^\circ \times \frac{\pi}{180} = \frac{125\pi}{180} = \frac{25\pi}{36}
\][/tex]
2. Determine the range within which the angle in radians falls:
We need to compare [tex]\(\frac{25\pi}{36}\)[/tex] to the boundary values of the specified ranges:
- The first range is from [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex]:
- [tex]\(\frac{\pi}{2} = \frac{180^\circ}{2} \times \frac{\pi}{180} = \frac{\pi}{2}\)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex]
- [tex]\(\frac{25\pi}{36} \approx 2.1817\)[/tex]
Comparing [tex]\(\frac{25\pi}{36}\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex], we conclude that [tex]\(\frac{25\pi}{36} > \frac{\pi}{2}\)[/tex].
- The second range is from [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex]:
- [tex]\(\pi = 180^\circ \times \frac{\pi}{180} = \pi\)[/tex]
- [tex]\(\pi \approx 3.1416\)[/tex]
Comparing [tex]\(\frac{25\pi}{36}\)[/tex] to [tex]\(\pi\)[/tex], we conclude that [tex]\(\frac{\pi}{2} < \frac{25\pi}{36} < \pi\)[/tex].
- Since [tex]\(\frac{25\pi}{36}\)[/tex] does not need to be compared to the third and fourth ranges (because it already falls within the specified ranges in second comparison), we can stop here.
Thus, when we convert [tex]\(125^\circ\)[/tex] to radians, the measure of the central angle in radians falls within the range:
[tex]\[
\boxed{\frac{\pi}{2} \text{ to } \pi}
\][/tex]
