To determine the measure of the central angle in radians and identify its range, we need to follow a few steps:
1. Convert Degrees to Radians:
The arc measure provided is [tex]\(295^\circ\)[/tex]. To convert degrees to radians, we use the conversion factor [tex]\(\frac{\pi \text{ radians}}{180^\circ}\)[/tex].
[tex]\[
\text{Angle in radians} = 295^\circ \times \frac{\pi \text{ radians}}{180^\circ}
\][/tex]
[tex]\[
\text{Angle in radians} \approx 5.149 \text{ radians}
\][/tex]
Therefore, the central angle [tex]\(295^\circ\)[/tex] is approximately [tex]\(5.149\)[/tex] radians.
2. Determine the Range for the Angle in Radians:
We need to compare the angle [tex]\(5.149\)[/tex] radians to the given ranges:
- [tex]\(0 \leq \theta < \frac{\pi}{2}\)[/tex] radians
- [tex]\(\frac{\pi}{2} \leq \theta < \pi\)[/tex] radians
- [tex]\(\pi \leq \theta < \frac{3\pi}{2}\)[/tex] radians
- [tex]\(\frac{3\pi}{2} \leq \theta \leq 2\pi\)[/tex] radians
First, we evaluate the boundaries of these ranges in radians:
[tex]\[
\frac{\pi}{2} \approx 1.571 \text{ radians}
\][/tex]
[tex]\[
\pi \approx 3.142 \text{ radians}
\][/tex]
[tex]\[
\frac{3\pi}{2} \approx 4.712 \text{ radians}
\][/tex]
[tex]\[
2\pi \approx 6.283 \text{ radians}
\][/tex]
Given [tex]\(5.149\)[/tex] radians, we see that it does not fall within the first three ranges.
The value [tex]\(5.149\)[/tex] radians fits within the final range:
[tex]\[
\frac{3\pi}{2} \leq 5.149 \leq 2\pi
\][/tex]
Therefore, the measure of the central angle in radians, approximately [tex]\(5.149\)[/tex], falls within the range [tex]\(\frac{3\pi}{2} \text{ to } 2\pi\)[/tex] radians.
