Convert $120^{\circ}$ To Radians.A. $\frac{2 \pi}{3}$ Radians B. $ 5 Π 6 \frac{5 \pi}{6} 6 5 Π ​ [/tex] Radians C. $\frac{3 \pi}{2}$ Radians D. $\frac{3 \pi}{4}$ Radians

Introduction

In mathematics, angles can be measured in two different units: degrees and radians. While degrees are commonly used in everyday life, radians are the standard unit of measurement in mathematics and physics. In this article, we will focus on converting degrees to radians, with a specific example of converting $120^{\circ}$ to radians.

Understanding Degrees and Radians

Degrees are a unit of measurement for angles, with 360 degrees in a full circle. Radians, on the other hand, are a unit of measurement for angles, with $2\pi$ radians in a full circle. To convert degrees to radians, we use the following formula:

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

Converting $120^{\circ}$ to Radians

Now, let's apply the formula to convert $120^{\circ}$ to radians.

$$\text{radians} = \frac{\pi}{180} \times 120$$

$$\text{radians} = \frac{120\pi}{180}$$

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

Therefore, $120^{\circ}$ is equal to $\frac{2\pi}{3}$ radians.

Why is this Conversion Important?

Converting degrees to radians is an essential skill in mathematics and physics. In calculus, for example, we often work with trigonometric functions, which are typically defined in terms of radians. By converting degrees to radians, we can ensure that our calculations are accurate and consistent.

Real-World Applications

Converting degrees to radians has numerous real-world applications. In engineering, for example, we often work with circular motion and rotation, which require precise calculations of angles in radians. In navigation, converting degrees to radians is essential for calculating distances and directions.

Conclusion

In conclusion, converting degrees to radians is a fundamental skill in mathematics and physics. By understanding the relationship between degrees and radians, we can perform accurate calculations and solve complex problems. In this article, we have demonstrated how to convert $120^{\circ}$ to radians using the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$. We hope that this article has provided a comprehensive guide to converting degrees to radians and has inspired you to explore the world of mathematics and physics.

Common Mistakes to Avoid

When converting degrees to radians, it's essential to avoid common mistakes. Here are a few tips to help you avoid errors:

• Make sure to use the correct formula: $\text{radians} = \frac{\pi}{180} \times \text{degrees}$
• Double-check your calculations: ensure that you have performed the correct arithmetic operations.
• Use a calculator or computer software to verify your results.

Practice Problems

To reinforce your understanding of converting degrees to radians, try the following practice problems:

1. Convert $90^{\circ}$ to radians.
2. Convert $270^{\circ}$ to radians.
3. Convert $360^{\circ}$ to radians.

Solutions

1. $$\text{radians} = \frac{\pi}{180} \times 90 = \frac{\pi}{2}$$

2. $$\text{radians} = \frac{\pi}{180} \times 270 = \frac{3\pi}{2}$$

3. $$\text{radians} = \frac{\pi}{180} \times 360 = 2\pi$$

Conclusion

Q&A: Converting Degrees to Radians

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

A: The formula for converting degrees to radians is:

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

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

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

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

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

Therefore, $180^{\circ}$ is equal to $\pi$ radians.

Q: What is the relationship between degrees and radians?

A: The relationship between degrees and radians is that $360^{\circ}$ is equal to $2\pi$ radians. This means that there are $\frac{2\pi}{360} = \frac{\pi}{180}$ radians in one degree.

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

A: To convert a negative angle from degrees to radians, we use the same formula as before:

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

However, we must also take into account the sign of the angle. If the angle is negative, we will get a negative value in radians.

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 for converting degrees to radians. Simply enter the angle in degrees and press the "radians" button to get the result.

Q: What are some common mistakes to avoid when converting degrees to radians?

A: Some common mistakes to avoid when converting degrees to radians include:

• Forgetting to use the correct formula
• Not double-checking calculations
• Not taking into account the sign of the angle
• Not using a calculator or computer software to verify results

Q: How do I convert radians to degrees?

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

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

Q: What are some real-world applications of converting degrees to radians?

A: Some real-world applications of converting degrees to radians include:

• Calculating distances and directions in navigation
• Working with circular motion and rotation in engineering
• Performing calculations in calculus and physics

Q: Can I use a computer software to convert degrees to radians?

A: Yes, you can use a computer software to convert degrees to radians. Most computer software, such as MATLAB or Python, have built-in functions for converting degrees to radians.

Conclusion

In conclusion, converting degrees to radians is a fundamental skill in mathematics and physics. By understanding the relationship between degrees and radians, we can perform accurate calculations and solve complex problems. We hope that this article has provided a comprehensive guide to converting degrees to radians and has inspired you to explore the world of mathematics and physics.