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Sagot :
To determine the measure of the central angle in radians for an arc that measures [tex]\(125^\circ\)[/tex], we need to convert the angle from degrees to radians and then evaluate which range it falls into.
### Step-by-Step Solution:
1. Convert degrees to radians:
- We start with the given angle in degrees: [tex]\(125^\circ\)[/tex].
- To convert degrees to radians, we use the conversion factor [tex]\(\frac{\pi}{180}\)[/tex] radians per degree.
[tex]\[ \text{Angle in radians} = 125^\circ \times \frac{\pi}{180} \][/tex]
2. Calculate the angle in radians:
- Performing the conversion:
[tex]\[ 125^\circ \times \frac{\pi}{180} = \frac{125\pi}{180} \][/tex]
Simplify the fraction [tex]\(\frac{125}{180}\)[/tex]:
[tex]\[ \frac{125}{180} = \frac{25}{36} \][/tex]
So, the angle in radians is:
[tex]\[ \frac{25\pi}{36} \][/tex]
3. Determine the numerical value of the angle in radians:
- Let's evaluate the numerical value of [tex]\(\frac{25\pi}{36}\)[/tex]:
[tex]\[ \frac{25\pi}{36} \approx 2.181661564992912 \, \text{radians} \][/tex]
4. Evaluate which range the central angle falls into:
- The given radian ranges are:
- From [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians ([tex]\(0\)[/tex] to approx. 1.5708 radians)
- From [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians (approx. 1.5708 to 3.1416 radians)
- From [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians (approx. 3.1416 to 4.7124 radians)
- From [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians (approx. 4.7124 to 6.2832 radians)
5. Place the calculated angle in the correct range:
- The calculated radian measure is approximately 2.1817 radians.
- Comparing this with the given ranges, we see that:
[tex]\[ \frac{\pi}{2} \leq 2.1817 < \pi \][/tex]
- Since [tex]\(\frac{\pi}{2}\)[/tex] is approximately 1.5708 and [tex]\(\pi\)[/tex] is approximately 3.1416, the angle [tex]\(2.1817\)[/tex] radians falls within the range:
[tex]\[ \frac{\pi}{2} \leq \text{angle} < \pi \][/tex]
Thus, the measure of the central angle in radians falls within the range:
[tex]\[ \boxed{\frac{\pi}{2} \text{ to } \pi \text{ radians}} \][/tex]
### Step-by-Step Solution:
1. Convert degrees to radians:
- We start with the given angle in degrees: [tex]\(125^\circ\)[/tex].
- To convert degrees to radians, we use the conversion factor [tex]\(\frac{\pi}{180}\)[/tex] radians per degree.
[tex]\[ \text{Angle in radians} = 125^\circ \times \frac{\pi}{180} \][/tex]
2. Calculate the angle in radians:
- Performing the conversion:
[tex]\[ 125^\circ \times \frac{\pi}{180} = \frac{125\pi}{180} \][/tex]
Simplify the fraction [tex]\(\frac{125}{180}\)[/tex]:
[tex]\[ \frac{125}{180} = \frac{25}{36} \][/tex]
So, the angle in radians is:
[tex]\[ \frac{25\pi}{36} \][/tex]
3. Determine the numerical value of the angle in radians:
- Let's evaluate the numerical value of [tex]\(\frac{25\pi}{36}\)[/tex]:
[tex]\[ \frac{25\pi}{36} \approx 2.181661564992912 \, \text{radians} \][/tex]
4. Evaluate which range the central angle falls into:
- The given radian ranges are:
- From [tex]\(0\)[/tex] to [tex]\(\frac{\pi}{2}\)[/tex] radians ([tex]\(0\)[/tex] to approx. 1.5708 radians)
- From [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians (approx. 1.5708 to 3.1416 radians)
- From [tex]\(\pi\)[/tex] to [tex]\(\frac{3\pi}{2}\)[/tex] radians (approx. 3.1416 to 4.7124 radians)
- From [tex]\(\frac{3\pi}{2}\)[/tex] to [tex]\(2\pi\)[/tex] radians (approx. 4.7124 to 6.2832 radians)
5. Place the calculated angle in the correct range:
- The calculated radian measure is approximately 2.1817 radians.
- Comparing this with the given ranges, we see that:
[tex]\[ \frac{\pi}{2} \leq 2.1817 < \pi \][/tex]
- Since [tex]\(\frac{\pi}{2}\)[/tex] is approximately 1.5708 and [tex]\(\pi\)[/tex] is approximately 3.1416, the angle [tex]\(2.1817\)[/tex] radians falls within the range:
[tex]\[ \frac{\pi}{2} \leq \text{angle} < \pi \][/tex]
Thus, the measure of the central angle in radians falls within the range:
[tex]\[ \boxed{\frac{\pi}{2} \text{ to } \pi \text{ radians}} \][/tex]
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