An Arc On A Circle Measures ${ 85^{\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.

Introduction

In mathematics, particularly in geometry and trigonometry, angles are measured in two different units: degrees and radians. While degrees are commonly used in everyday applications, radians are the standard unit of measurement in mathematics and physics. In this article, we will explore the relationship between degrees and radians, and how to convert between these two units.

The Measure of an Angle in Degrees and Radians

An angle is measured in degrees, denoted by the symbol °, and is a unit of measurement for angles. The measure of an angle in degrees is a number that represents the amount of rotation from the initial side to the terminal side of the angle. On the other hand, an angle is measured in radians, denoted by the symbol rad, and is a unit of measurement for angles in terms of the radius of a circle.

Converting Degrees to Radians

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

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

This formula states that to convert an angle from degrees to radians, we multiply the angle in degrees by $\frac{\pi}{180}$.

Converting Radians to Degrees

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

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

This formula states that to convert an angle from radians to degrees, we multiply the angle in radians by $\frac{180}{\pi}$.

The Measure of the Central Angle in Radians

Now, let's consider the problem at hand. We are given that an arc on a circle measures $85^{\circ}$. We need to find the measure of the central angle in radians.

Step 1: Convert the Measure of the Arc from Degrees to Radians

Using the formula for converting degrees to radians, we have:

$$\text{radians} = \frac{\pi}{180} \times 85^{\circ}$$

$$\text{radians} = \frac{\pi}{180} \times \frac{85}{1}$$

$$\text{radians} = \frac{85\pi}{180}$$

Step 2: Simplify the Expression

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

$$\text{radians} = \frac{85\pi}{180}$$

$$\text{radians} = \frac{17\pi}{36}$$

Step 3: Determine the Range of the Central Angle in Radians

Now that we have the measure of the central angle in radians, we need to determine the range in which it lies.

Range A: $0$ to $\frac{\pi}{2}$ radians

To determine if the central angle lies in this range, we need to compare its measure with the upper limit of the range, which is $\frac{\pi}{2}$ radians.

$$\frac{17\pi}{36} > \frac{\pi}{2}$$

$$\frac{17\pi}{36} \approx 1.48$$

$$\frac{\pi}{2} \approx 1.57$$

Since $\frac{17\pi}{36}$ is greater than $\frac{\pi}{2}$, the central angle does not lie in this range.

Range B: $\frac{\pi}{2}$ to $\pi$ radians

To determine if the central angle lies in this range, we need to compare its measure with the lower limit of the range, which is $\frac{\pi}{2}$ radians.

$$\frac{17\pi}{36} > \frac{\pi}{2}$$

$$\frac{17\pi}{36} \approx 1.48$$

$$\frac{\pi}{2} \approx 1.57$$

Since $\frac{17\pi}{36}$ is greater than $\frac{\pi}{2}$, the central angle does not lie in this range.

Conclusion

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

Final Answer

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Q: What is the relationship between degrees and radians?

A: Degrees and radians are two different units of measurement for angles. While degrees are commonly used in everyday applications, radians are the standard unit of measurement in mathematics and physics.

Q: How do I convert degrees to radians?

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

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

Q: How do I convert radians to degrees?

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

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

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

A: The measure of the central angle in radians is given by the formula:

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

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

A: To determine the range of the central angle in radians, you need to compare its measure with the upper and lower limits of the range. If the measure of the central angle is greater than the upper limit, it does not lie in that range. If the measure of the central angle is less than the lower limit, it does not lie in that range.

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

A: The range of the central angle in radians is $\frac{\pi}{2}$ to $\pi$ radians.

Q: How do I simplify the expression for the measure of the central angle in radians?

A: To simplify the expression for the measure of the central angle in radians, you can divide both the numerator and the denominator by their greatest common divisor.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{B}$.

Common Mistakes to Avoid

• Not converting the measure of the arc from degrees to radians: Make sure to convert the measure of the arc from degrees to radians before finding the measure of the central angle in radians.
• Not simplifying the expression for the measure of the central angle in radians: Make sure to simplify the expression for the measure of the central angle in radians by dividing both the numerator and the denominator by their greatest common divisor.
• Not determining the range of the central angle in radians: Make sure to determine the range of the central angle in radians by comparing its measure with the upper and lower limits of the range.

Additional Resources

• Mathematics textbooks: Consult a mathematics textbook for more information on the relationship between degrees and radians.
• Online resources: Visit online resources such as Khan Academy, Mathway, or Wolfram Alpha for more information on the relationship between degrees and radians.
• Practice problems: Practice solving problems involving the relationship between degrees and radians to improve your understanding and skills.