An Arc On A Circle Measures $295^{\circ}$. The Measure Of The Central Angle, In Radians, Is Within Which Range?A. $0$ To $\frac{\pi}{2}$ Radians B. $\frac{\pi}{2}$ To $\pi$ Radians C. $\pi$ To

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 commonly used in everyday life, while radians are used in mathematical and scientific applications. To convert degrees to radians, we use the following formula:

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

Converting $295^{\circ}$ to Radians

Using the formula above, we can convert $295^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 295$$

$$\text{radians} = \frac{295\pi}{180}$$

$$\text{radians} = \frac{59\pi}{36}$$

Understanding the Range of Radians

To determine the range in which the measure of the central angle falls, we need to understand the relationship between radians and the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The angle $\theta$ in radians is measured counterclockwise from the positive x-axis.

Ranges of Radians

The range of radians can be divided into four quadrants:

• Quadrant I: $0$ to $\frac{\pi}{2}$ radians
• Quadrant II: $\frac{\pi}{2}$ to $\pi$ radians
• Quadrant III: $\pi$ to $\frac{3\pi}{2}$ radians
• Quadrant IV: $\frac{3\pi}{2}$ to $2\pi$ radians

Determining the Range of the Central Angle

To determine the range in which the measure of the central angle falls, we need to compare the converted radians to the ranges above.

$$\frac{59\pi}{36}$$

is greater than $\pi$ and less than $\frac{3\pi}{2}$.

Conclusion

In conclusion, the measure of the central angle, in radians, is within the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Final Answer

The final answer is $\boxed{C}$

Introduction

In our previous article, we discussed how to convert a central angle from degrees to radians and determine the range in which the measure of the central angle falls. In this article, we will address some frequently asked questions (FAQs) related to this topic.

Q&A

Q1: What is the relationship between degrees and radians?

A1: Degrees and radians are two units used to measure angles. Degrees are commonly used in everyday life, while radians are used in mathematical and scientific applications. To convert degrees to radians, we use the formula:

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

Q2: How do I convert a central angle from degrees to radians?

A2: To convert a central angle from degrees to radians, you can use the formula above. For example, to convert $295^{\circ}$ to radians, you would use:

$$\text{radians} = \frac{\pi}{180} \times 295$$

$$\text{radians} = \frac{295\pi}{180}$$

$$\text{radians} = \frac{59\pi}{36}$$

Q3: What is the range of radians?

A3: The range of radians can be divided into four quadrants:

• Quadrant I: $0$ to $\frac{\pi}{2}$ radians
• Quadrant II: $\frac{\pi}{2}$ to $\pi$ radians
• Quadrant III: $\pi$ to $\frac{3\pi}{2}$ radians
• Quadrant IV: $\frac{3\pi}{2}$ to $2\pi$ radians

Q4: How do I determine the range of the central angle?

A4: To determine the range of the central angle, you need to compare the converted radians to the ranges above. For example, if the converted radians is $\frac{59\pi}{36}$, it falls within the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Q5: What is the final answer for the central angle $295^{\circ}$?

A5: The final answer for the central angle $295^{\circ}$ is $\boxed{C}$, which corresponds to the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Conclusion

In conclusion, we have addressed some frequently asked questions related to converting central angles from degrees to radians. We hope this article has provided you with a better understanding of the topic and has helped you to determine the range of the central angle.

Final Tips

• Always use the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ to convert degrees to radians.
• Compare the converted radians to the ranges above to determine the range of the central angle.
• Practice converting central angles from degrees to radians to become more comfortable with the process.

Final Answer

The final answer is $\boxed{C}$.