To determine the range in which the central angle falls, we need to first convert the given measure of the central angle from degrees to radians and then analyze its range.
1. Understanding the Given Information:
- The given central angle measures [tex]\(85^{\circ}\)[/tex].
2. Conversion from Degrees to Radians:
- To convert degrees to radians, we use the conversion factor [tex]\(\pi \ \text{radians} = 180^{\circ}\)[/tex].
- Hence, [tex]\( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \left( \frac{\pi}{180} \right) \)[/tex].
Using this formula, we compute:
[tex]\[
\theta_{\text{radians}} = 85^{\circ} \times \left( \frac{\pi}{180} \right).
\][/tex]
Substituting the values:
[tex]\[
\theta_{\text{radians}} = 85 \times \left( \frac{\pi}{180} \right) \approx 1.4835298641951802 \ \text{radians}.
\][/tex]
3. Determining the Range of the Central Angle:
- We will now determine the range within which this central angle in radians falls. The options are:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians,
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians,
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians,
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians.
We know:
- [tex]\(\frac{\pi}{2} \approx 1.57079632679\)[/tex],
- [tex]\(\pi \approx 3.14159265359\)[/tex],
- [tex]\(\frac{3\pi}{2} \approx 4.71238898038\)[/tex],
- [tex]\(2\pi \approx 6.28318530718\)[/tex].
Comparing [tex]\(1.4835298641951802\)[/tex] radians:
- [tex]\(0 \leq 1.4835298641951802 < \frac{\pi}{2} \approx 1.57079632679\)[/tex].
Therefore, the central angle of [tex]\(85^{\circ}\)[/tex] (which converts to roughly [tex]\(1.4835298641951802\)[/tex] radians) falls within the range:
[tex]\[
\boxed{0 \ \text{to} \ \frac{\pi}{2} \ \text{radians}}.
\][/tex]
