An Arc On A Circle Measures 125 ∘ 125^{\circ} 12 5 ∘ . The Measure Of The Central Angle, In Radians, Is Within Which Range?A. 0 To Π 2 \frac{\pi}{2} 2 Π ​ Radians B. Π 2 \frac{\pi}{2} 2 Π ​ To Π \pi Π Radians C. Π \pi Π To

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Introduction

When dealing with circles and angles, it's essential to understand the relationship between degrees and radians. In this article, we will explore how to convert a central angle from degrees to radians and determine the range in which the measure of the central angle falls.

Understanding Degrees and Radians

Degrees and radians are two units used to measure angles. Degrees are a more common unit, but radians are often used in mathematical calculations, especially when dealing with trigonometric functions.

To convert degrees to radians, we use the following formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Converting $125^{\circ}$ to Radians

Using the formula above, we can convert $125^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 125$$

$$\text{radians} = \frac{125\pi}{180}$$

Simplifying the Fraction

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5:

$$\text{radians} = \frac{25\pi}{36}$$

Understanding the Range of Radians

Now that we have the measure of the central angle in radians, we need to determine the range in which it falls. The options given are:

A. 0 to $\frac{\pi}{2}$ radians B. $\frac{\pi}{2}$ to $\pi$ radians C. $\pi$ to $2\pi$ radians

Comparing the Measure of the Central Angle to the Options

To determine the range in which the measure of the central angle falls, we need to compare it to the options given. We can do this by converting the options to decimal form:

A. 0 to $\frac{\pi}{2}$ radians = 0 to 1.57 radians B. $\frac{\pi}{2}$ to $\pi$ radians = 1.57 to 3.14 radians C. $\pi$ to $2\pi$ radians = 3.14 to 6.28 radians

Determining the Range

Now that we have the options in decimal form, we can compare them to the measure of the central angle:

$$\text{radians} = \frac{25\pi}{36} \approx 2.19$$

Since 2.19 is greater than 1.57 and less than 3.14, the measure of the central angle falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Conclusion

In this article, we explored how to convert a central angle from degrees to radians and determined the range in which the measure of the central angle falls. We used the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ to convert $125^{\circ}$ to radians and simplified the fraction to $\frac{25\pi}{36}$. We then compared the measure of the central angle to the options given and determined that it falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Final Answer

The final answer is B. $\frac{\pi}{2}$ to $\pi$ radians.

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Introduction

In our previous article, we explored how to convert a central angle from degrees to radians and determined the range in which the measure of the central angle falls. In this Q&A article, we will answer some common questions related to the topic.

Q: What is the relationship between degrees and radians?

A: Degrees and radians are two units used to measure angles. Degrees are a more common unit, but radians are often used in mathematical calculations, especially when dealing with trigonometric functions.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the following formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q: What is the measure of the central angle in radians for an arc of $125^{\circ}$?

A: Using the formula above, we can convert $125^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 125$$

$$\text{radians} = \frac{125\pi}{180}$$

$$\text{radians} = \frac{25\pi}{36}$$

Q: What is the range in which the measure of the central angle falls?

A: Since $\frac{25\pi}{36} \approx 2.19$, the measure of the central angle falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Q: Why is it important to understand the relationship between degrees and radians?

A: Understanding the relationship between degrees and radians is essential in mathematics, especially when dealing with trigonometric functions. It allows us to convert between different units of measurement and perform calculations more easily.

Q: Can I use a calculator to convert degrees to radians?

A: Yes, you can use a calculator to convert degrees to radians. Simply enter the degree value and press the "radians" button or use the formula above.

Q: What are some common applications of radians in real-life scenarios?

A: Radians are used in various real-life scenarios, such as:

• Calculating distances and angles in navigation and geography
• Determining the area and circumference of circles in engineering and architecture
• Analyzing data and making predictions in statistics and data science

Q: Can I use radians to measure angles in everyday life?

A: While radians are not commonly used in everyday life, they are essential in mathematical calculations and are used in various fields such as engineering, physics, and computer science.

Conclusion

In this Q&A article, we answered some common questions related to the topic of converting degrees to radians and determining the range in which the measure of the central angle falls. We hope that this article has provided you with a better understanding of the relationship between degrees and radians and how to apply it in real-life scenarios.

Final Answer

The final answer is B. $\frac{\pi}{2}$ to $\pi$ radians.