Convert The Angle $\theta = 260^{\circ}$ To Radians. Express Your Answer Exactly. $\theta = $ _____ Radians

Introduction

In mathematics, angles are often measured in degrees or radians. While degrees are commonly used in everyday applications, radians are the standard unit of measurement in calculus and other advanced mathematical disciplines. In this article, we will explore the process of converting angles from degrees to radians, with a focus on the conversion of the angle $\theta = 260^{\circ}$.

Understanding Degrees and Radians

Before we dive into the conversion process, it's essential to understand the difference between degrees and radians. A degree is a unit of measurement for angles, with 360 degrees in a full circle. On the other hand, a radian is a unit of measurement for angles, with the circumference of a circle being equal to $2\pi$ radians.

The Conversion Formula

The conversion formula from degrees to radians is given by:

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

This formula can be used to convert any angle from degrees to radians.

Converting $\theta = 260^{\circ}$ to Radians

Now that we have the conversion formula, let's apply it to the angle $\theta = 260^{\circ}$. Plugging in the value of $\theta$ into the formula, we get:

$$\text{radians} = \frac{\pi}{180} \times 260^{\circ}$$

To simplify the expression, we can multiply the numerator and denominator by 10:

$$\text{radians} = \frac{10\pi}{1800} \times 260^{\circ}$$

Now, we can simplify the expression further by canceling out the common factor of 10:

$$\text{radians} = \frac{\pi}{180} \times 26^{\circ}$$

Since we want to express the answer exactly, we can use the fact that $\pi$ is an irrational number. Therefore, we can write the answer as:

$$\text{radians} = \frac{26\pi}{180}$$

Simplifying the Answer

To simplify the answer further, we can divide both the numerator and denominator by their greatest common divisor, which is 2:

$$\text{radians} = \frac{13\pi}{90}$$

Conclusion

In this article, we have explored the process of converting angles from degrees to radians. We have applied the conversion formula to the angle $\theta = 260^{\circ}$ and simplified the answer to get the final result of $\frac{13\pi}{90}$ radians.

Key Takeaways

• The conversion formula from degrees to radians is given by: $\text{radians} = \frac{\pi}{180} \times \text{degrees}$
• To convert an angle from degrees to radians, we can use the conversion formula and simplify the expression to get the final result.
• The answer should be expressed exactly, using irrational numbers such as $\pi$.

Additional Resources

For more information on converting angles from degrees to radians, check out the following resources:

• Khan Academy: Converting between degrees and radians
• Mathway: Converting degrees to radians
• Wolfram Alpha: Converting degrees to radians

Frequently Asked Questions

Q: What is the conversion formula from degrees to radians? A: The conversion formula from degrees to radians is given by: $\text{radians} = \frac{\pi}{180} \times \text{degrees}$

Q: How do I simplify the answer after converting an angle from degrees to radians? A: To simplify the answer, you can divide both the numerator and denominator by their greatest common divisor.

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Q&A: Converting Degrees to Radians

Q: What is the conversion formula from degrees to radians?

A: The conversion formula from degrees to radians is given by:

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

Q: How do I convert an angle from degrees to radians?

A: To convert an angle from degrees to radians, you can use the conversion formula and simplify the expression to get the final result.

Q: What is the difference between degrees and radians?

A: A degree is a unit of measurement for angles, with 360 degrees in a full circle. On the other hand, a radian is a unit of measurement for angles, with the circumference of a circle being equal to $2\pi$ radians.

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

A: Yes, you can use a calculator to convert degrees to radians. However, it's essential to understand the conversion formula and be able to simplify the expression to get the final result.

Q: How do I simplify the answer after converting an angle from degrees to radians?

A: To simplify the answer, you can divide both the numerator and denominator by their greatest common divisor.

Q: What is the final result of converting $\theta = 260^{\circ}$ to radians?

A: The final result is $\frac{13\pi}{90}$ radians.

Q: Can I use the conversion formula to convert an angle from radians to degrees?

A: Yes, you can use the conversion formula to convert an angle from radians to degrees. The formula is given by:

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

Q: How do I choose between using degrees and radians?

A: The choice between using degrees and radians depends on the context and the specific problem you are trying to solve. In general, radians are used in calculus and other advanced mathematical disciplines, while degrees are used in everyday applications.

Q: Can I use the conversion formula to convert an angle from degrees to radians in a specific quadrant?

A: Yes, you can use the conversion formula to convert an angle from degrees to radians in a specific quadrant. However, you need to be aware of the quadrant and adjust the angle accordingly.

Q: What is the relationship between degrees and radians in a full circle?

A: In a full circle, there are 360 degrees and $2\pi$ radians.

Q: Can I use the conversion formula to convert an angle from degrees to radians in a partial circle?

A: Yes, you can use the conversion formula to convert an angle from degrees to radians in a partial circle. However, you need to be aware of the partial circle and adjust the angle accordingly.

Q: How do I convert an angle from degrees to radians in a 3D space?

A: To convert an angle from degrees to radians in a 3D space, you need to use the conversion formula and take into account the orientation of the angle in the 3D space.

Q: Can I use the conversion formula to convert an angle from degrees to radians in a spherical coordinate system?

A: Yes, you can use the conversion formula to convert an angle from degrees to radians in a spherical coordinate system. However, you need to be aware of the spherical coordinate system and adjust the angle accordingly.

Q: What is the relationship between degrees and radians in a spherical coordinate system?

A: In a spherical coordinate system, there are 360 degrees and $2\pi$ radians in a full circle.

Q: Can I use the conversion formula to convert an angle from degrees to radians in a cylindrical coordinate system?

A: Yes, you can use the conversion formula to convert an angle from degrees to radians in a cylindrical coordinate system. However, you need to be aware of the cylindrical coordinate system and adjust the angle accordingly.

Q: How do I convert an angle from degrees to radians in a cylindrical coordinate system?

A: To convert an angle from degrees to radians in a cylindrical coordinate system, you need to use the conversion formula and take into account the orientation of the angle in the cylindrical coordinate system.

Q: Can I use the conversion formula to convert an angle from degrees to radians in a polar coordinate system?

A: Yes, you can use the conversion formula to convert an angle from degrees to radians in a polar coordinate system. However, you need to be aware of the polar coordinate system and adjust the angle accordingly.

Q: How do I convert an angle from degrees to radians in a polar coordinate system?

A: To convert an angle from degrees to radians in a polar coordinate system, you need to use the conversion formula and take into account the orientation of the angle in the polar coordinate system.

Conclusion

In this article, we have explored the process of converting angles from degrees to radians. We have answered common questions and provided examples to help you understand the conversion formula and simplify the answer. Whether you are a student or a professional, understanding the conversion formula and being able to simplify the answer is essential for working with angles in mathematics and science.