Answer :
To determine the range of the central angle of an arc that measures [tex]\(295^\circ\)[/tex] when converted to radians, follow these steps:
1. Convert angles from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[ \text{angle in radians} = \text{angle in degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
- Applying this formula:
[tex]\[ 295^\circ \times \left(\frac{\pi}{180}\right) = 5.1487212933832724 \text{ radians} \][/tex]
2. Determine the range of the angle in radians:
- We need to see within which of the following ranges [tex]\(5.1487212933832724\)[/tex] radians falls:
1. [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
2. [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
3. [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians
4. [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians
3. Evaluate the boundaries of these ranges:
[tex]\[ \begin{align*} 0 & \leq \text{angle} < \frac{\pi}{2} \quad \text{(approximately } 0 \leq \text{angle} < 1.5708) \\ \frac{\pi}{2} & \leq \text{angle} < \pi \quad \text{(approximately } 1.5708 \leq \text{angle} < 3.1416) \\ \pi & \leq \text{angle} < \frac{3\pi}{2} \quad \text{(approximately } 3.1416 \leq \text{angle} < 4.7124) \\ \frac{3\pi}{2} & \leq \text{angle} \leq 2\pi \quad \text{(approximately } 4.7124 \leq \text{angle} \leq 6.2832) \\ \end{align*} \][/tex]
4. Compare the angle [tex]\(5.1487212933832724\)[/tex] radians with these ranges:
[tex]\(5.1487212933832724\)[/tex] radians is greater than [tex]\(4.7124\)[/tex] but less than [tex]\(6.2832\)[/tex].
Therefore, the central angle of [tex]\(295^\circ\)[/tex] (or [tex]\(5.1487212933832724\)[/tex] radians) falls within the range:
[tex]\[ \frac{3\pi}{2} \text{ to } 2\pi \text{ radians} \][/tex]
1. Convert angles from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[ \text{angle in radians} = \text{angle in degrees} \times \left(\frac{\pi}{180}\right) \][/tex]
- Applying this formula:
[tex]\[ 295^\circ \times \left(\frac{\pi}{180}\right) = 5.1487212933832724 \text{ radians} \][/tex]
2. Determine the range of the angle in radians:
- We need to see within which of the following ranges [tex]\(5.1487212933832724\)[/tex] radians falls:
1. [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
2. [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians
3. [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians
4. [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians
3. Evaluate the boundaries of these ranges:
[tex]\[ \begin{align*} 0 & \leq \text{angle} < \frac{\pi}{2} \quad \text{(approximately } 0 \leq \text{angle} < 1.5708) \\ \frac{\pi}{2} & \leq \text{angle} < \pi \quad \text{(approximately } 1.5708 \leq \text{angle} < 3.1416) \\ \pi & \leq \text{angle} < \frac{3\pi}{2} \quad \text{(approximately } 3.1416 \leq \text{angle} < 4.7124) \\ \frac{3\pi}{2} & \leq \text{angle} \leq 2\pi \quad \text{(approximately } 4.7124 \leq \text{angle} \leq 6.2832) \\ \end{align*} \][/tex]
4. Compare the angle [tex]\(5.1487212933832724\)[/tex] radians with these ranges:
[tex]\(5.1487212933832724\)[/tex] radians is greater than [tex]\(4.7124\)[/tex] but less than [tex]\(6.2832\)[/tex].
Therefore, the central angle of [tex]\(295^\circ\)[/tex] (or [tex]\(5.1487212933832724\)[/tex] radians) falls within the range:
[tex]\[ \frac{3\pi}{2} \text{ to } 2\pi \text{ radians} \][/tex]
