First, we need to convert the given angle from degrees to radians. The formula to convert degrees to radians is:
[tex]\[ \text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
Given the angle:
[tex]\[ 250^\circ \][/tex]
Let's convert this to radians:
[tex]\[ 250^\circ \times \left(\frac{\pi}{180}\right) = \frac{250\pi}{180} = \frac{25\pi}{18} \][/tex]
Now, calculate the approximate value:
[tex]\[ \frac{25\pi}{18} \approx 4.363 \text{ radians} \][/tex]
Next, we need to determine within which range this radian measure falls.
We have four specified ranges:
1. [tex]\( 0 \text{ to } \frac{\pi}{2} \)[/tex] radians
2. [tex]\( \frac{\pi}{2} \text{ to } \pi \)[/tex] radians
3. [tex]\( \pi \text{ to } \frac{3\pi}{2} \)[/tex] radians
4. [tex]\( \frac{3\pi}{2} \text{ to } 2\pi \)[/tex] radians
Let's evaluate these ranges using their approximations:
1. [tex]\( 0 \text{ to } \frac{\pi}{2} \)[/tex]: [tex]\( 0 \text{ to } 1.5708 \)[/tex]
2. [tex]\( \frac{\pi}{2} \text{ to } \pi \)[/tex]: [tex]\( 1.5708 \text{ to } 3.1416 \)[/tex]
3. [tex]\( \pi \text{ to } \frac{3\pi}{2} \)[/tex]: [tex]\( 3.1416 \text{ to } 4.7124 \)[/tex]
4. [tex]\( \frac{3\pi}{2} \text{ to } 2\pi \)[/tex]: [tex]\( 4.7124 \text{ to } 6.2832 \)[/tex]
The radian measure [tex]\( 4.363 \)[/tex] falls into the third range:
[tex]\[ \pi \text{ ranges to } \frac{3\pi}{2} \approx 3.1416 \text{ to } 4.7124 \][/tex]
Thus, the central angle of 250 degrees in radians is approximately 4.363, placing it within the range [tex]\( \pi \text{ to } \frac{3\pi}{2} \)[/tex] radians.
