To determine the range within which the measure of the central angle in radians lies, given an arc measure of [tex]\( 85^{\circ} \)[/tex], we need to perform the following steps:
1. Convert the degrees to radians:
The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
Applying this formula to [tex]\( 85^{\circ} \)[/tex]:
[tex]\[
85^{\circ} \times \frac{\pi}{180} = \frac{85\pi}{180}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{85\pi}{180} = \frac{17\pi}{36}
\][/tex]
2. Determine the range of [tex]\(\frac{17\pi}{36}\)[/tex]:
We need to check which range the value [tex]\(\frac{17\pi}{36}\)[/tex] falls into:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians:
[tex]\[
0 \le \text{radians} < \frac{\pi}{2}
\][/tex]
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians:
[tex]\[
\frac{\pi}{2} \le \text{radians} < \pi
\][/tex]
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians:
[tex]\[
\pi \le \text{radians} < \frac{3\pi}{2}
\][/tex]
- [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians:
[tex]\[
\frac{3\pi}{2} \le \text{radians} \le 2\pi
\][/tex]
We now need to compute [tex]\(\frac{\pi}{2}\)[/tex] in terms of a fraction with denominator 36 to make the comparison easier:
[tex]\[
\frac{\pi}{2} = \frac{18\pi}{36}
\][/tex]
Comparing [tex]\(\frac{17\pi}{36}\)[/tex] with [tex]\(\frac{18\pi}{36}\)[/tex]:
[tex]\[
\frac{17\pi}{36} < \frac{18\pi}{36}
\][/tex]
Since [tex]\(\frac{17\pi}{36} < \frac{\pi}{2}\)[/tex], [tex]\(\frac{17\pi}{36}\)[/tex] lies within the range:
[tex]\[
0 \le \text{radians} < \frac{\pi}{2}
\][/tex]
Therefore, the measure of the central angle [tex]\(85^{\circ}\)[/tex] in radians falls in the range: [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians.
