To determine the range in which the measure of the central angle falls, we need to convert the given angle from degrees to radians and then identify its range.
1. Convert the angle from degrees to radians:
The angle given is [tex]\( 295^\circ \)[/tex].
To convert this angle to radians, we use the conversion factor [tex]\( \pi \)[/tex] radians [tex]\( = 180^\circ \)[/tex].
The formula to convert degrees to radians is:
[tex]\[
\text{angle\_radians} = \text{angle\_degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Substituting [tex]\( 295^\circ \)[/tex] into the formula:
[tex]\[
\text{angle\_radians} = 295 \times \left( \frac{\pi}{180} \right) \approx 5.1487212933832724
\][/tex]
2. Determine the range in which the angle in radians falls:
- The range from [tex]\( 0 \)[/tex] to [tex]\( \frac{\pi}{2} \)[/tex] radians is approximately 0 to 1.5708 radians.
- The range from [tex]\( \frac{\pi}{2} \)[/tex] to [tex]\( \pi \)[/tex] radians is approximately 1.5708 to 3.1416 radians.
- The range from [tex]\( \pi \)[/tex] to [tex]\( \frac{3\pi}{2} \)[/tex] radians is approximately 3.1416 to 4.7124 radians.
- The range from [tex]\( \frac{3\pi}{2} \)[/tex] to [tex]\( 2\pi \)[/tex] radians is approximately 4.7124 to 6.2832 radians.
The calculated angle in radians is approximately 5.1487.
By comparing this value with the given ranges:
[tex]\[
\pi \approx 3.1416 \quad \text{and} \quad \frac{3\pi}{2} \approx 4.7124
\][/tex]
We see that:
[tex]\[
4.7124 < 5.1487 < 6.2832
\][/tex]
Therefore, the angle in radians falls within the range from [tex]\( \frac{3\pi}{2} \)[/tex] to [tex]\( 2\pi \)[/tex] radians.
Hence, the measure of the central angle, [tex]\( 295^\circ \)[/tex], converted to radians is approximately [tex]\( 5.1487 \)[/tex] radians and it falls within the range [tex]\( \frac{3\pi}{2} \)[/tex] to [tex]\( 2\pi \)[/tex] radians.
