To determine the range of the central angle that corresponds to an arc measuring [tex]\( 125^\circ \)[/tex], we need to perform a few steps:
1. Convert the angle from degrees to radians:
The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
For our given angle of [tex]\( 125^\circ \)[/tex]:
[tex]\[
125^\circ \times \frac{\pi}{180} = \frac{125\pi}{180} \approx 2.182 \text{ radians}
\][/tex]
2. Determine the range of the angle in radians:
Let's check which interval the angle [tex]\( 2.182 \)[/tex] radians falls into.
- Check [tex]\( 0 < 2.182 \leq \frac{\pi}{2} \)[/tex]:
[tex]\(\frac{\pi}{2} \approx 1.57\)[/tex], and since [tex]\(2.182 > 1.57\)[/tex], this interval does not include [tex]\(2.182\)[/tex].
- Check [tex]\(\frac{\pi}{2} < 2.182 \leq \pi\)[/tex]:
[tex]\(\pi \approx 3.14\)[/tex], and since [tex]\(1.57 < 2.182 < 3.14\)[/tex], this interval does include [tex]\(2.182\)[/tex].
- Check [tex]\(\pi < 2.182 \leq \frac{3\pi}{2}\)[/tex]:
[tex]\(\frac{3\pi}{2} \approx 4.71\)[/tex], and since [tex]\(2.182 < \pi (3.14)\)[/tex], this interval does not include [tex]\(2.182\)[/tex].
- Check [tex]\(\frac{3\pi}{2} < 2.182 \leq 2\pi\)[/tex]:
[tex]\(2\pi \approx 6.28\)[/tex], and since [tex]\(2.182 < 3\pi/2 (4.71)\)[/tex], this interval does not include [tex]\(2.182\)[/tex].
Therefore, the central angle in radians, which is approximately [tex]\(2.182\)[/tex] radians, falls within the range [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians.
