To solve this problem, let's follow these steps:
1. Convert the angle from degrees to radians.
- The arc measure given is [tex]\(85^\circ\)[/tex].
- To convert degrees to radians, use the conversion factor [tex]\(\frac{\pi}{180}\)[/tex].
So, we have:
[tex]\[
\text{Central angle in radians} = 85 \times \frac{\pi}{180}
\][/tex]
2. Calculate the central angle in radians.
- Multiplying the degrees by the conversion factor:
[tex]\[
85 \times \frac{\pi}{180} = \frac{85\pi}{180} \approx 1.4835298641951802 \text{ radians}
\][/tex]
3. Determine which range the central angle in radians falls within.
- We have the following ranges to consider:
- [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
- The value [tex]\(1.4835298641951802\)[/tex] radians lies between [tex]\(0\)[/tex] and [tex]\(\frac{\pi}{2}\)[/tex].
- Since [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex], and [tex]\(1.4835298641951802 < 1.5708\)[/tex], the central angle falls within the first range.
4. Conclusion:
- The central angle, [tex]\(1.4835298641951802\)[/tex] radians, falls within the range [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians.
Therefore, the central angle is within the range [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2} \)[/tex] radians.
