To determine which range the measure of the central angle falls in, we need to follow these steps:
1. Convert the Angle from Degrees to Radians:
- The given angle is [tex]\(85^\circ\)[/tex].
- To convert degrees to radians, use the conversion factor: [tex]\(1^\circ = \frac{\pi}{180}\)[/tex] radians.
- Therefore, [tex]\(85^\circ\)[/tex] can be converted to radians as follows:
[tex]\[
85 \times \frac{\pi}{180} = \frac{85\pi}{180}
\][/tex]
- Simplifying this, we get:
[tex]\[
\frac{85\pi}{180} = \frac{17\pi}{36}
\][/tex]
- In decimal form, approximately:
[tex]\[
\frac{17\pi}{36} \approx 1.4835298641951802 \text{ radians}
\][/tex]
2. Determine the Range:
- We need to identify the range in which [tex]\(1.4835298641951802\)[/tex] radians lies.
- The given ranges are:
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
- Let's compare [tex]\(1.4835298641951802\)[/tex] radians with the boundary values of these ranges.
- [tex]\(\frac{\pi}{2} \approx 1.5707963267948966\)[/tex] radians
- [tex]\(\pi \approx 3.141592653589793\)[/tex] radians
- [tex]\(\frac{3\pi}{2} \approx 4.71238898038469\)[/tex] radians
- [tex]\(2\pi \approx 6.283185307179586\)[/tex] radians
- Clearly,
[tex]\[
0 \leq 1.4835298641951802 < 1.5707963267948966
\][/tex]
This shows that [tex]\(1.4835298641951802\)[/tex] radians falls in the first range.
Therefore, the measure of the central angle in radians lies within the range:
[tex]\[0 \text{ to } \frac{\pi}{2} \text{ radians}\][/tex]
Answer: 0 to [tex]\(\frac{\pi}{2}\)[/tex] radians.
