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 $\frac{3

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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 trigonometry and circular 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:

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

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

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

Determining the Range

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

The standard ranges for radians are:

• $0$ to $\frac{\pi}{2}$ radians
• $\frac{\pi}{2}$ to $\pi$ radians
• $\pi$ to $\frac{3\pi}{2}$ radians
• $\frac{3\pi}{2}$ to $2\pi$ radians

Comparing the Converted Value

Now, let's compare the converted value $\frac{25\pi}{36}$ to the standard ranges:

• $\frac{25\pi}{36}$ is greater than $0$ and less than $\frac{\pi}{2}$, but it's also greater than $\frac{\pi}{4}$.
• $\frac{25\pi}{36}$ is less than $\pi$ and greater than $\frac{3\pi}{4}$.

Conclusion

Based on the comparison, we can conclude that the measure of the central angle, in radians, is within the range of $\frac{\pi}{4}$ to $\pi$ radians.

Final Answer

The final answer is $\boxed{\frac{\pi}{4} \text{ to } \pi}$ radians.

Discussion

This problem requires a good understanding of the relationship between degrees and radians. It also requires the ability to compare values and determine the correct range.

Additional Tips

• When converting degrees to radians, make sure to use the correct formula.
• When comparing values, make sure to consider the standard ranges for radians.
• Practice converting degrees to radians and comparing values to become more comfortable with the concept.

Related Topics

• Converting degrees to radians
• Comparing values
• Standard ranges for radians

References

• [1] "Degrees and Radians" by Math Open Reference
• [2] "Converting Degrees to Radians" by Khan Academy
• [3] "Standard Ranges for Radians" by Wolfram MathWorld
  

Introduction

In our previous article, we explored 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 answer some frequently asked questions (FAQs) on converting degrees to radians.

Q&A

Q1: What is the formula for converting degrees to radians?

A1: The formula for converting degrees to radians is:

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

Q2: How do I convert $180^{\circ}$ to radians?

A2: To convert $180^{\circ}$ to radians, we use the formula:

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

$$\text{radians} = \pi$$

Q3: What is the range of values for radians?

A3: The standard ranges for radians are:

• $0$ to $\frac{\pi}{2}$ radians
• $\frac{\pi}{2}$ to $\pi$ radians
• $\pi$ to $\frac{3\pi}{2}$ radians
• $\frac{3\pi}{2}$ to $2\pi$ radians

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

A4: To determine the range of a central angle in radians, we need to compare the converted value to the standard ranges for radians.

Q5: What is the difference between degrees and radians?

A5: 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 trigonometry and circular functions.

Q6: Can I convert radians to degrees?

A6: Yes, you can convert radians to degrees using the following formula:

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

Q7: How do I convert a mixed angle from degrees to radians?

A7: To convert a mixed angle from degrees to radians, we need to convert the whole number part and the fractional part separately.

For example, to convert $45^{\circ} 30^{\prime}$ to radians, we first convert the whole number part:

$$\text{radians} = \frac{\pi}{180} \times 45$$

$$\text{radians} = \frac{\pi}{4}$$

Then, we convert the fractional part:

$$\text{radians} = \frac{\pi}{180} \times 30$$

$$\text{radians} = \frac{\pi}{6}$$

Finally, we add the two values together:

$$\text{radians} = \frac{\pi}{4} + \frac{\pi}{6}$$

$$\text{radians} = \frac{3\pi}{12} + \frac{2\pi}{12}$$

$$\text{radians} = \frac{5\pi}{12}$$

Conclusion

Converting degrees to radians is an essential skill in mathematics, especially when dealing with trigonometry and circular functions. By understanding the formula and the standard ranges for radians, you can easily convert degrees to radians and determine the range of a central angle.

Final Tips

• Practice converting degrees to radians and comparing values to become more comfortable with the concept.
• Use the correct formula to convert degrees to radians.
• Consider the standard ranges for radians when determining the range of a central angle.

Related Topics

• Converting degrees to radians
• Comparing values
• Standard ranges for radians
• Converting radians to degrees
• Mixed angles

References

• [1] "Degrees and Radians" by Math Open Reference
• [2] "Converting Degrees to Radians" by Khan Academy
• [3] "Standard Ranges for Radians" by Wolfram MathWorld