An Arc On A Circle Measures $125^{\circ}$. The Measure Of The Central Angle, In Radians, Is Within Which Range?A. $0$ To $\frac{\pi}{2}$ RadiansB. $\frac{\pi}{2}$ To $\pi$ RadiansC. $\pi$ To

Introduction

When working with circles and angles, it's essential to understand the relationship between degrees and radians. This article will explore how to convert degrees to radians and apply this knowledge to a specific problem involving a central angle on a circle.

Degrees and Radians: A Brief Overview

Degrees and radians are two units used to measure angles. Degrees are a more common unit, often used in everyday applications, while radians are commonly used in mathematics and trigonometry.

• Degrees: A degree is a unit of measurement for angles, with 360 degrees in a full circle.
• Radians: A radian is a unit of measurement for angles, with 2π radians in a full circle.

Converting Degrees to Radians

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

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

This formula allows us to convert any angle measured in degrees to radians.

The Problem: An Arc on a Circle Measures $125^{\circ}$

We are given that an arc on a circle measures $125^{\circ}$. We need to find the measure of the central angle in radians.

Finding the Central Angle in Radians

To find the central angle in radians, we can use the formula:

$$\text{central angle in radians} = \text{arc length in degrees} \times \frac{\pi}{180}$$

Plugging in the given value, we get:

$$\text{central angle in radians} = 125^{\circ} \times \frac{\pi}{180}$$

Simplifying the Expression

To simplify the expression, we can multiply the numerator and denominator by 1:

$$\text{central angle in radians} = \frac{125\pi}{180}$$

Reducing the Fraction

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

$$\text{central angle in radians} = \frac{25\pi}{36}$$

Evaluating the Expression

To evaluate the expression, we can multiply the numerator and denominator by 1:

$$\text{central angle in radians} = \frac{25\pi}{36}$$

Determining the Range of the Central Angle

To determine the range of the central angle, we need to compare it to the values in the given options.

• Option A: $0$ to $\frac{\pi}{2}$ radians
• Option B: $\frac{\pi}{2}$ to $\pi$ radians
• Option C: $\pi$ to $2\pi$ radians

Comparing the Central Angle to the Options

To compare the central angle to the options, we can evaluate the expression:

$$\frac{25\pi}{36}$$

This value is greater than $\frac{\pi}{2}$ but less than $\pi$.

Conclusion

Based on the comparison, we can conclude that the measure of the central angle, in radians, is within the range of $\frac{\pi}{2}$ to $\pi$ radians.

The final answer is: $\boxed{B}$

Introduction

In the previous article, we explored the relationship between degrees and radians, and how to convert degrees to radians. In this article, we will answer some frequently asked questions related to degrees and radians.

Q: What is the difference between degrees and radians?

A: Degrees and radians are two units used to measure angles. Degrees are a more common unit, often used in everyday applications, while radians are commonly used in mathematics and trigonometry.

Q: How do I convert degrees to radians?

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

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

Q: How do I convert radians to degrees?

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

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

Q: What is the relationship between degrees and radians?

A: The relationship between degrees and radians is that 360 degrees is equal to 2π radians.

Q: How do I find the central angle in radians?

A: To find the central angle in radians, you can use the formula:

$$\text{central angle in radians} = \text{arc length in degrees} \times \frac{\pi}{180}$$

Q: What is the range of the central angle in radians?

A: The range of the central angle in radians depends on the value of the arc length in degrees. If the arc length is between 0 and 180 degrees, the central angle is between 0 and π radians. If the arc length is between 180 and 360 degrees, the central angle is between π and 2π radians.

Q: Can I use degrees and radians interchangeably?

A: No, you cannot use degrees and radians interchangeably. While they are related, they have different units and are used in different contexts.

Q: Why are radians used in mathematics and trigonometry?

A: Radians are used in mathematics and trigonometry because they provide a more precise and consistent way of measuring angles. They are also used in many mathematical formulas and equations.

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

A: Yes, you can use a calculator to convert degrees to radians. Most calculators have a built-in function to convert degrees to radians.

Conclusion

In this article, we answered some frequently asked questions related to degrees and radians. We hope this article has provided you with a better understanding of the relationship between degrees and radians, and how to convert between them.

Frequently Asked Questions: Degrees and Radians

• Q: What is the difference between degrees and radians? A: Degrees and radians are two units used to measure angles. Degrees are a more common unit, often used in everyday applications, while radians are commonly used in mathematics and trigonometry.
• Q: How do I convert degrees to radians? A: To convert degrees to radians, you can use the following formula:

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

• Q: How do I convert radians to degrees? A: To convert radians to degrees, you can use the following formula:

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

• Q: What is the relationship between degrees and radians? A: The relationship between degrees and radians is that 360 degrees is equal to 2π radians.

• Q: How do I find the central angle in radians? A: To find the central angle in radians, you can use the formula:

$$\text{central angle in radians} = \text{arc length in degrees} \times \frac{\pi}{180}$$

• Q: What is the range of the central angle in radians? A: The range of the central angle in radians depends on the value of the arc length in degrees. If the arc length is between 0 and 180 degrees, the central angle is between 0 and π radians. If the arc length is between 180 and 360 degrees, the central angle is between π and 2π radians.

• Q: Can I use degrees and radians interchangeably? A: No, you cannot use degrees and radians interchangeably. While they are related, they have different units and are used in different contexts.

• Q: Why are radians used in mathematics and trigonometry? A: Radians are used in mathematics and trigonometry because they provide a more precise and consistent way of measuring angles. They are also used in many mathematical formulas and equations.

• Q: Can I use a calculator to convert degrees to radians? A: Yes, you can use a calculator to convert degrees to radians. Most calculators have a built-in function to convert degrees to radians.