To determine within which range the arc measure of [tex]\(295^\circ\)[/tex] falls in the context of radians, we need to follow these steps:
1. Convert the arc measure from degrees to radians:
To convert degrees to radians, use the formula:
[tex]\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\][/tex]
For an arc measure of [tex]\(295^\circ\)[/tex]:
[tex]\[
295^\circ \times \frac{\pi}{180} = \frac{295\pi}{180} \approx 5.1487212933832724 \text{ radians}
\][/tex]
2. Determine the range in which the central angle falls:
The possible ranges for the central angle, in radians, are:
- [tex]\(0 \text{ to } \frac{\pi}{2} \text{ radians}\)[/tex]
- [tex]\(\frac{\pi}{2} \text{ to } \pi \text{ radians}\)[/tex]
- [tex]\(\pi \text{ to } \frac{3\pi}{2} \text{ radians}\)[/tex]
- [tex]\(\frac{3\pi}{2} \text{ to } 2\pi \text{ radians}\)[/tex]
Let's convert these to approximate decimal values for easier comparison:
- [tex]\(\frac{\pi}{2} \approx 1.5708\)[/tex]
- [tex]\(\pi \approx 3.1416\)[/tex]
- [tex]\(\frac{3\pi}{2} \approx 4.7124\)[/tex]
- [tex]\(2\pi \approx 6.2832\)[/tex]
Now, we find that:
- [tex]\(0\)[/tex] to [tex]\(1.5708\)[/tex] radians does not include [tex]\(5.1487\)[/tex] radians
- [tex]\(1.5708\)[/tex] to [tex]\(3.1416\)[/tex] radians does not include [tex]\(5.1487\)[/tex] radians
- [tex]\(3.1416\)[/tex] to [tex]\(4.7124\)[/tex] radians does not include [tex]\(5.1487\)[/tex] radians
- [tex]\(4.7124\)[/tex] to [tex]\(6.2832\)[/tex] radians does include [tex]\(5.1487\)[/tex] radians
Therefore, the arc measure of [tex]\(295^\circ\)[/tex] falls within the range [tex]\(\frac{3\pi}{2} \text{ to } 2\pi\)[/tex] radians. So, the measure of the central angle, when expressed in radians, is in the range:
[tex]\[
\boxed{\frac{3\pi}{2} \text{ to } 2\pi \text{ radians}}
\][/tex]
