To determine the measure of the central angle in radians and its range, follow the steps below:
1. Convert the angle from degrees to radians:
- We know that [tex]\(1 \text{ degree} = \frac{\pi}{180} \text{ radians}\)[/tex].
- Therefore, to convert [tex]\(85^\circ\)[/tex] to radians, we use the conversion factor:
[tex]\[
85^\circ \times \frac{\pi}{180} = \frac{85\pi}{180}
\][/tex]
- Simplify [tex]\(\frac{85\pi}{180}\)[/tex]:
[tex]\[
\frac{85\pi}{180} = \frac{17\pi}{36} \approx 1.4835298641951802 \text{ radians}
\][/tex]
2. Determine the range within which the angle in radians falls:
- We compare [tex]\(1.4835298641951802\)[/tex] radians with the given boundaries:
- [tex]\(0 \text{ to } \frac{\pi}{2}\)[/tex] radians
- [tex]\(\frac{\pi}{2} \text{ to } \pi\)[/tex] radians
- [tex]\(\pi \text{ to } \frac{3\pi}{2}\)[/tex] radians
- [tex]\(\frac{3\pi}{2} \text{ to } 2\pi\)[/tex] radians
- Knowing that [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex] and [tex]\(\pi \approx 3.1416\)[/tex], we can see that the value [tex]\(1.4835298641951802\)[/tex] radians falls within:
[tex]\[
0 \text{ to } \frac{\pi}{2} \text{ radians} \quad \text{(since 0 ≤ 1.4835298641951802 < 1.5708)}
\][/tex]
So, the measure of the central angle [tex]\(85^\circ\)[/tex] in radians is approximately [tex]\(1.4835298641951802\)[/tex], and it falls within the range [tex]\(0 \text{ to } \frac{\pi}{2} \text{ radians}\)[/tex].
