To determine the measure of the central angle in radians and identify its range, follow these steps:
### Step 1: Convert the Angle from Degrees to Radians
Given the central angle measures [tex]\( 295^\circ \)[/tex] in degrees, we first need to convert this measurement to radians. The conversion factor between degrees and radians is:
[tex]\[
1^\circ = \frac{\pi}{180} \text{ radians}
\][/tex]
So, to convert [tex]\( 295^\circ \)[/tex] to radians, we use the formula:
[tex]\[
\text{angle\_rad} = 295 \times \left(\frac{\pi}{180}\right)
\][/tex]
### Step 2: Calculate the Exact Radian Measure
The calculation gives us:
[tex]\[
295 \times \left(\frac{\pi}{180}\right) = 5.1487212933832724 \text{ radians}
\][/tex]
### Step 3: Determine the Range in Radians
Now, we need to determine which of the provided ranges this angle falls into:
1. [tex]\( 0 \leq \theta < \frac{\pi}{2} \)[/tex]
2. [tex]\( \frac{\pi}{2} \leq \theta < \pi \)[/tex]
3. [tex]\( \pi \leq \theta < \frac{3\pi}{2} \)[/tex]
4. [tex]\( \frac{3\pi}{2} \leq \theta < 2\pi \)[/tex]
Substitute the known value [tex]\( \theta = 5.1487212933832724 \text{ radians} \)[/tex]:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2} \approx 0 < \theta < 1.571 \text{ radians} \)[/tex]
- [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi \approx 1.571 \leq \theta < 3.142 \text{ radians} \)[/tex]
- [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2} \approx 3.142 \leq \theta < 4.712 \text{ radians} \ )
- \(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi \approx 4.712 \leq \theta < 6.283 \text{ radians} \)[/tex]
Since [tex]\(5.1487212933832724 > 4.712\)[/tex], it is clear that [tex]\(5.1487212933832724 < 6.283 \)[/tex].
Therefore, [tex]\(5.1487212933832724\)[/tex] radians falls into the range:
[tex]\[
\boxed{\frac{3\pi}{2} \text{ to } 2\pi}
\][/tex]
This means the angle is within the range:
[tex]\[
\boxed{\frac{3\pi}{2} \text{ to } 2\pi \text{ radians}}
\][/tex]
