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To find out the range within which a central angle, measured in radians, falls given an arc measurement of [tex]\(85^{\circ}\)[/tex], let's convert the angle from degrees to radians and then determine its range.
### Step-by-Step Solution:
1. Convert the angle from degrees to radians:
- The conversion factor from degrees to radians is [tex]\(\frac{\pi}{180}\)[/tex].
- So, the angle in radians is calculated as:
[tex]\[ \text{Angle in radians} = 85^{\circ} \times \left(\frac{\pi}{180}\right) \][/tex]
2. Calculate the angle in radians:
- Plugging in the value, we get:
[tex]\[ 85^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{85\pi}{180} \approx 1.4835298641951802 \, \text{radians} \][/tex]
3. Evaluate which range this angle falls into:
- We know the ranges are defined as:
- [tex]\(0 \, \text{to} \, \frac{\pi}{2} \, \text{radians}\)[/tex]
- [tex]\(\frac{\pi}{2} \, \text{to} \, \pi \, \text{radians}\)[/tex]
- [tex]\(\pi \, \text{to} \, \frac{3\pi}{2} \, \text{radians}\)[/tex]
- [tex]\(\frac{3\pi}{2} \, \text{to} \, 2\pi \, \text{radians}\)[/tex]
- Now, let's compare our result of approximately [tex]\(1.4835298641951802 \, \text{radians}\)[/tex] to these ranges:
- [tex]\(0 \leq 1.4835 < \frac{\pi}{2} = 1.5708\)[/tex]
- [tex]\(\frac{\pi}{2} = 1.5708 \leq 1.4835 < \pi = 3.1416\)[/tex]
- [tex]\(\pi = 3.1416 \leq 1.4835 < \frac{3\pi}{2} = 4.7124\)[/tex]
- [tex]\(\frac{3\pi}{2} = 4.7124 \leq 1.4835 < 2\pi = 6.2832\)[/tex]
- From our comparisons, [tex]\(1.4835298641951802\)[/tex] radians falls within the range of [tex]\(0 \, \text{to} \, \frac{\pi}{2}\)[/tex] radians.
### Conclusion:
The central angle measuring [tex]\(85^{\circ}\)[/tex] is approximately [tex]\(1.4835\)[/tex] radians, which falls into the range from [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians. Thus, the correct range option is:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
### Step-by-Step Solution:
1. Convert the angle from degrees to radians:
- The conversion factor from degrees to radians is [tex]\(\frac{\pi}{180}\)[/tex].
- So, the angle in radians is calculated as:
[tex]\[ \text{Angle in radians} = 85^{\circ} \times \left(\frac{\pi}{180}\right) \][/tex]
2. Calculate the angle in radians:
- Plugging in the value, we get:
[tex]\[ 85^{\circ} \times \left(\frac{\pi}{180}\right) = \frac{85\pi}{180} \approx 1.4835298641951802 \, \text{radians} \][/tex]
3. Evaluate which range this angle falls into:
- We know the ranges are defined as:
- [tex]\(0 \, \text{to} \, \frac{\pi}{2} \, \text{radians}\)[/tex]
- [tex]\(\frac{\pi}{2} \, \text{to} \, \pi \, \text{radians}\)[/tex]
- [tex]\(\pi \, \text{to} \, \frac{3\pi}{2} \, \text{radians}\)[/tex]
- [tex]\(\frac{3\pi}{2} \, \text{to} \, 2\pi \, \text{radians}\)[/tex]
- Now, let's compare our result of approximately [tex]\(1.4835298641951802 \, \text{radians}\)[/tex] to these ranges:
- [tex]\(0 \leq 1.4835 < \frac{\pi}{2} = 1.5708\)[/tex]
- [tex]\(\frac{\pi}{2} = 1.5708 \leq 1.4835 < \pi = 3.1416\)[/tex]
- [tex]\(\pi = 3.1416 \leq 1.4835 < \frac{3\pi}{2} = 4.7124\)[/tex]
- [tex]\(\frac{3\pi}{2} = 4.7124 \leq 1.4835 < 2\pi = 6.2832\)[/tex]
- From our comparisons, [tex]\(1.4835298641951802\)[/tex] radians falls within the range of [tex]\(0 \, \text{to} \, \frac{\pi}{2}\)[/tex] radians.
### Conclusion:
The central angle measuring [tex]\(85^{\circ}\)[/tex] is approximately [tex]\(1.4835\)[/tex] radians, which falls into the range from [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians. Thus, the correct range option is:
- [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians
Answer: Â Choice A
Explanation
[tex]\frac{\pi}{2}\ \text{radians} = 90^{\circ}[/tex] is something you should memorize since it comes up a lot in trigonometry.
As an alternative, you can multiply [tex]\frac{\pi}{2}[/tex] by the conversion factor [tex]\frac{180}{\pi}[/tex] to arrive at 90°
Since 0 < 85 < 90, it means the central angle 85° is between 0 radians and [tex]\frac{\pi}{2}[/tex] radians.
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