To determine the range in which the measure of a central angle in radians falls, given an arc measures [tex]\(85^\circ\)[/tex], we can follow these steps:
1. Convert the angle from degrees to radians:
The formula to convert degrees to radians is:
[tex]\[
\text{angle in radians} = \left(\frac{\text{angle in degrees} \times \pi}{180}\right)
\][/tex]
Given the angle is [tex]\(85^\circ\)[/tex]:
[tex]\[
\text{angle in radians} = \left(\frac{85 \times \pi}{180}\right)
\][/tex]
Simplifying this expression:
[tex]\[
\text{angle in radians} = \left(\frac{85}{180}\right) \pi \approx 1.4835298641951802 \text{ radians}
\][/tex]
2. Determine the range category for the angle in radians:
We need to check within which of the provided ranges [tex]\(1.4835298641951802\)[/tex] radians falls:
- [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex] radians: This range is approximately 0 to 1.5708 radians.
- [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex] radians: This range is approximately 1.5708 to 3.1416 radians.
- [tex]\(\pi \leq \theta < \frac{3\pi}{2}\)[/tex] radians: This range is approximately 3.1416 to 4.7124 radians.
- [tex]\(\frac{3\pi}{2} \leq \theta < 2\pi\)[/tex] radians: This range is approximately 4.7124 to 6.2832 radians.
Comparing [tex]\(1.4835298641951802\)[/tex] radians with these ranges:
- It is not within the range [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex] radians because [tex]\(1.4835298641951802\)[/tex] is slightly less than [tex]\(\frac{\pi}{2}\)[/tex] radians.
- It falls within the range [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex] radians as [tex]\(1.4835298641951802\)[/tex] is less than [tex]\(\pi\)[/tex] but more than [tex]\(\frac{\pi}{2}\)[/tex].
Therefore, the measure of the central angle [tex]\(85^\circ\)[/tex] in radians falls within the range:
[tex]\[
0 \leq \theta < \frac{\pi}{2} \text{ radians}
\][/tex]
