An Arc On A Circle Measures $250^{\circ}$. Within Which Range Is The Radian Measure Of The Central Angle?A. 0 To $\frac{\pi}{2}$ Radians B. $ Π 2 \frac{\pi}{2} 2 Π ​ [/tex] To $\pi$ Radians C. $\pi$ To

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Understanding the Relationship Between Degrees and Radians

To determine the range of the radian measure of the central angle, we need to understand the relationship between degrees and radians. A full circle is equal to $360^{\circ}$, and it is also equal to $2\pi$ radians. This relationship can be used to convert between degrees and radians.

Converting Degrees to Radians

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

radians=π180×degrees\text{radians} = \frac{\pi}{180} \times \text{degrees}

Using this formula, we can convert $250^{\circ}$ to radians:

radians=π180×250\text{radians} = \frac{\pi}{180} \times 250

radians=250π180\text{radians} = \frac{250\pi}{180}

radians=25π18\text{radians} = \frac{25\pi}{18}

Understanding the Range of Radian Measures

Now that we have the radian measure of the central angle, we need to determine the range within which it falls. The range of radian measures can be broken down into several intervals:

  • 0 to $\frac{\pi}{2}$ radians: This range corresponds to the first quadrant of the unit circle, where the angle is acute.

  • \frac{\pi}{2}$ to $\pi$ radians: This range corresponds to the second quadrant of the unit circle, where the angle is obtuse.

  • \pi$ to $\frac{3\pi}{2}$ radians: This range corresponds to the third quadrant of the unit circle, where the angle is obtuse.

  • \frac{3\pi}{2}$ to $2\pi$ radians: This range corresponds to the fourth quadrant of the unit circle, where the angle is acute.

Determining the Range of the Radian Measure

Now that we have the radian measure of the central angle, we can determine the range within which it falls. The radian measure is $\frac{25\pi}{18}$, which is greater than $\frac{3\pi}{2}$ and less than $2\pi$. Therefore, the radian measure of the central angle falls within the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Conclusion

In conclusion, the radian measure of the central angle that corresponds to an arc on a circle measuring $250^{\circ}$ falls within the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Frequently Asked Questions

  • What is the relationship between degrees and radians?
  • How do I convert degrees to radians?
  • What is the range of radian measures?
  • How do I determine the range of the radian measure of the central angle?

Final Answer

The final answer is: π\boxed{\pi}

Understanding the Relationship Between Degrees and Radians

To determine the range of the radian measure of the central angle, we need to understand the relationship between degrees and radians. A full circle is equal to $360^{\circ}$, and it is also equal to $2\pi$ radians. This relationship can be used to convert between degrees and radians.

Q: What is the relationship between degrees and radians?

A: A full circle is equal to $360^{\circ}$, and it is also equal to $2\pi$ radians. This relationship can be used to convert between degrees and radians.

Q: How do I convert degrees to radians?

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

radians=π180×degrees\text{radians} = \frac{\pi}{180} \times \text{degrees}

Q: What is the range of radian measures?

A: The range of radian measures can be broken down into several intervals:

  • 0 to $\frac{\pi}{2}$ radians: This range corresponds to the first quadrant of the unit circle, where the angle is acute.

  • \frac{\pi}{2}$ to $\pi$ radians: This range corresponds to the second quadrant of the unit circle, where the angle is obtuse.

  • \pi$ to $\frac{3\pi}{2}$ radians: This range corresponds to the third quadrant of the unit circle, where the angle is obtuse.

  • \frac{3\pi}{2}$ to $2\pi$ radians: This range corresponds to the fourth quadrant of the unit circle, where the angle is acute.

Q: How do I determine the range of the radian measure of the central angle?

A: To determine the range of the radian measure of the central angle, we need to convert the degree measure to radians and then determine the range within which it falls.

Q: What is the radian measure of the central angle that corresponds to an arc on a circle measuring $250^{\circ}$?

A: The radian measure of the central angle that corresponds to an arc on a circle measuring $250^{\circ}$ is $\frac{25\pi}{18}$.

Q: Within which range does the radian measure of the central angle fall?

A: The radian measure of the central angle falls within the range of $\pi$ to $\frac{3\pi}{2}$ radians.

Q: What is the final answer?

A: The final answer is: π\boxed{\pi}

Conclusion

In conclusion, understanding the relationship between degrees and radians is crucial in determining the range of the radian measure of the central angle. By using the formula to convert degrees to radians and determining the range within which the radian measure falls, we can accurately determine the range of the radian measure of the central angle.

Frequently Asked Questions

  • What is the relationship between degrees and radians?
  • How do I convert degrees to radians?
  • What is the range of radian measures?
  • How do I determine the range of the radian measure of the central angle?
  • What is the radian measure of the central angle that corresponds to an arc on a circle measuring $250^{\circ}$?
  • Within which range does the radian measure of the central angle fall?

Final Answer

The final answer is: π\boxed{\pi}