To determine the measure of the central angle in radians given that an arc on a circle measures [tex]\( 295^{\circ} \)[/tex], follow these steps:
1. Convert the angle from degrees to radians:
[tex]\[
\text{Angle in radians} = \text{Angle in degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
Substituting [tex]\( 295^{\circ} \)[/tex]:
[tex]\[
\text{Angle in radians} = 295 \times \left(\frac{\pi}{180}\right)
\][/tex]
[tex]\[
\text{Angle in radians} = \frac{295\pi}{180}
\][/tex]
Simplifying this, we get approximately:
[tex]\[
\text{Angle in radians} \approx 5.1487212933832724
\][/tex]
2. Determine the range in which this radian measure falls:
The given ranges for the central angle in radians are:
- [tex]\(0 \text{ to } \frac{\pi}{2}\)[/tex]
- [tex]\(\frac{\pi}{2} \text{ to } \pi\)[/tex]
- [tex]\(\pi \text{ to } \frac{3\pi}{2}\)[/tex]
- [tex]\(\frac{3\pi}{2} \text{ to } 2\pi\)[/tex]
First, convert these ranges to approximate decimal values using [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
\begin{align*}
0 \text{ to } \frac{\pi}{2} & \approx 0 \text{ to } 1.5708 \\
\frac{\pi}{2} \text{ to } \pi & \approx 1.5708 \text{ to } 3.1416 \\
\pi \text{ to } \frac{3\pi}{2} & \approx 3.1416 \text{ to } 4.7124 \\
\frac{3\pi}{2} \text{ to } 2\pi & \approx 4.7124 \text{ to } 6.2832 \\
\end{align*}
\][/tex]
Now we check where [tex]\( 5.1487212933832724 \)[/tex] fits:
- [tex]\(0 \text{ to } 1.5708\)[/tex]: Out of range
- [tex]\(1.5708 \text{ to } 3.1416\)[/tex]: Out of range
- [tex]\(3.1416 \text{ to } 4.7124\)[/tex]: Out of range
- [tex]\(4.7124 \text{ to } 6.2832\)[/tex]: This is the correct range
Thus, the central angle of the arc measuring [tex]\( 295^{\circ} \)[/tex] in radians falls within the range:
[tex]\(\frac{3\pi}{2} \text{ to } 2\pi\)[/tex].
