Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.