To determine the range of the radian measure of a central angle that measures [tex]\(250^{\circ}\)[/tex], we'll follow these steps:
1. Convert the angle from degrees to radians:
The radian measure can be found using the conversion factor, where [tex]\(1 \text{ degree} = \frac{\pi}{180}\)[/tex] radians.
[tex]\[
\text{Angle in radians} = 250^{\circ} \times \left( \frac{\pi}{180} \right)
\][/tex]
[tex]\[
\text{Angle in radians} = \frac{250 \pi}{180} = \frac{250 \pi}{180} = \frac{25 \pi}{18}
\][/tex]
2. Determine the numerical value in radians:
To understand the value in radians more precisely:
[tex]\[
\frac{25 \pi}{18} \approx 4.3633 \text{ radians}
\][/tex]
3. Identify the corresponding range:
- [tex]\(0 \; \text{radians} \leq \text{angle} < \frac{\pi}{2} \; \text{radians}\)[/tex] is approximately [tex]\(0 \text{ radians} \leq \text{angle} < 1.5708 \text{ radians}\)[/tex].
- [tex]\(\frac{\pi}{2} \; \text{radians} \leq \text{angle} < \pi \; \text{radians}\)[/tex] is approximately [tex]\(1.5708 \text{ radians} \leq \text{angle} < 3.1416 \text{ radians}\)[/tex].
- [tex]\(\pi \; \text{radians} \leq \text{angle} < \frac{3 \pi}{2} \; \text{radians}\)[/tex] is approximately [tex]\(3.1416 \text{ radians} \leq \text{angle} < 4.7124 \text{ radians}\)[/tex].
- [tex]\(\frac{3 \pi}{2} \; \text{radians} \leq \text{angle} < 2 \pi \; \text{radians}\)[/tex] is approximately [tex]\(4.7124 \text{ radians} \leq \text{angle} < 6.2832 \text{ radians}\)[/tex].
Given that [tex]\(4.3633 \text{ radians}\)[/tex] lies between [tex]\( \pi \; \text{radians} \approx 3.1416 \)[/tex] and [tex]\( \frac{3 \pi}{2} \; \text{radians} \approx 4.7124 \)[/tex], we see it falls in the range:
[tex]\[
\pi \; \text{radians} \leq \text{angle} < \frac{3 \pi}{2} \; \text{radians}
\][/tex]
Therefore, the radian measure of the central angle [tex]\(250^{\circ}\)[/tex] falls within the range:
\[
\pi \; \text{radians} \leq \text{angle} < \frac{3 \pi}{2} \; \text{radians}
\
