Certainly! Let's work through this step-by-step:
1. Understand the Given Value:
The measure of an arc is provided as [tex]\( 295^{\circ} \)[/tex].
2. Convert Degrees to Radians:
To convert degrees into radians, we use the conversion factor [tex]\( \pi \)[/tex] radians is [tex]\( 180^{\circ} \)[/tex].
[tex]\[
\text{Measure in radians} = \frac{295 \times \pi}{180}
\][/tex]
When you perform the calculation:
[tex]\[
\text{Measure in radians} \approx 5.1487212933832724 \text{ radians}
\][/tex]
3. Determine the Range:
Next, we need to determine which of the specified ranges the calculated radian measure falls into. The ranges are:
[tex]\[
0 \leq \theta < \frac{\pi}{2} \quad (\text{Quadrant I})
\][/tex]
[tex]\[
\frac{\pi}{2} \leq \theta < \pi \quad (\text{Quadrant II})
\][/tex]
[tex]\[
\pi \leq \theta < \frac{3\pi}{2} \quad (\text{Quadrant III})
\][/tex]
[tex]\[
\frac{3\pi}{2} \leq \theta < 2\pi \quad (\text{Quadrant IV})
\][/tex]
Knowing that [tex]\( \pi \approx 3.141592653589793 \)[/tex], let's calculate the bounds in radians:
[tex]\[
\frac{\pi}{2} \approx 1.5707963267948966
\][/tex]
[tex]\[
\pi \approx 3.141592653589793
\][/tex]
[tex]\[
\frac{3\pi}{2} \approx 4.71238898038469
\][/tex]
[tex]\[
2\pi \approx 6.283185307179586
\][/tex]
Now we compare our computed radian measure ([tex]\( 5.1487212933832724 \)[/tex]) with these bounds:
- Most importantly, [tex]\( 4.71238898038469 < 5.1487212933832724 < 6.283185307179586 \)[/tex].
This means the measure in radians falls into the fourth category [tex]\( \frac{3\pi}{2} \leq \theta < 2\pi \)[/tex].
4. Conclusion:
- The measure of the arc is [tex]\( 295^{\circ} \)[/tex].
- When converted to radians, it is approximately [tex]\( 5.1487212933832724 \)[/tex].
- This measure falls in the range of [tex]\( \frac{3\pi}{2} \leq \theta < 2\pi \)[/tex].
Thus, the arc measure in degrees and its equivalent in radians is [tex]\( 295^{\circ} \)[/tex] and [tex]\( 5.1487212933832724 \)[/tex] radians respectively, which places it in the fourth range [tex]\( \frac{3\pi}{2} \leq \theta < 2\pi \)[/tex].
