To determine within which range the measure of a central angle, measured at 125 degrees, falls in radians, follow these steps:
1. Convert the central angle from degrees to radians:
The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex]. Therefore, the central angle in radians can be found using:
[tex]\[
\text{Central Angle in Radians} = 125^\circ \times \frac{\pi}{180}
\][/tex]
2. Perform the calculation:
[tex]\[
125^\circ \times \frac{\pi}{180} = \frac{125 \pi}{180}
\][/tex]
Simplify this fraction:
[tex]\[
\frac{125 \pi}{180} = \frac{25 \pi}{36}
\][/tex]
This gives us the measure of the central angle in radians.
3. Determine the numerical value of the central angle in radians:
[tex]\[
\frac{25 \pi}{36} \approx 2.181661564992912 \, \text{radians}
\][/tex]
4. Identify which range the angle falls into:
Compare this value to the given ranges:
- [tex]\(0 \)[/tex] to [tex]\( \frac{\pi}{2} \approx 1.5708 \)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5708 \)[/tex] to [tex]\( \pi \approx 3.1416 \)[/tex]
- [tex]\(\pi \approx 3.1416 \)[/tex] to [tex]\( \frac{3\pi}{2} \approx 4.7124 \)[/tex]
- [tex]\(\frac{3\pi}{2} \approx 4.7124 \)[/tex] to [tex]\( 2\pi \approx 6.2832 \)[/tex]
So, [tex]\(2.181661564992912\)[/tex] falls between [tex]\( \frac{\pi}{2} \approx 1.5708 \)[/tex] and [tex]\( \pi \approx 3.1416 \)[/tex].
Hence, the measure of the central angle, in radians, falls in the range [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians.
