How Many Radians Is $-135^{\circ}$?A. $-\frac{1}{4} \pi$ B. $\frac{3}{4} \pi$ C. $-\frac{3}{4} \pi$ D. $-\frac{4}{3} \pi$

===========================================================

Understanding the Basics of Radian Measure

---

Radian measure is a fundamental concept in mathematics, particularly in trigonometry and calculus. It is essential to understand how to convert degrees to radians and vice versa. In this article, we will focus on converting a given angle from degrees to radians.

The Relationship Between Degrees and Radians

---

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

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

This formula is derived from the fact that a full circle is equal to $360^{\circ}$, which is also equal to $2\pi$ radians.

Converting $-135^{\circ}$ to Radians

---

Now, let's apply the formula to convert $-135^{\circ}$ to radians.

$$\text{radians} = \frac{\pi}{180} \times (-135^{\circ})$$

$$\text{radians} = \frac{\pi}{180} \times (-135)$$

$$\text{radians} = -\frac{135\pi}{180}$$

$$\text{radians} = -\frac{3\pi}{4}$$

Simplifying the Answer

---

We can simplify the answer by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$$\text{radians} = -\frac{3\pi}{4}$$

Conclusion

---

In conclusion, the correct answer is $-\frac{3}{4} \pi$. This is because we converted $-135^{\circ}$ to radians using the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ and simplified the answer to get the final result.

Common Mistakes to Avoid

---

When converting degrees to radians, it is essential to remember the following common mistakes:

• Not using the correct formula: Make sure to use the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ to convert degrees to radians.
• Not simplifying the answer: Simplify the answer by dividing both the numerator and the denominator by their greatest common divisor.
• Not checking the units: Make sure to check the units of the answer to ensure that it is in radians.

Real-World Applications

---

Converting degrees to radians has numerous real-world applications, including:

• Navigation: In navigation, angles are often measured in degrees, but calculations are often performed in radians.
• Engineering: In engineering, angles are often measured in degrees, but calculations are often performed in radians.
• Science: In science, angles are often measured in degrees, but calculations are often performed in radians.

Conclusion

---

In conclusion, converting degrees to radians is a fundamental concept in mathematics, particularly in trigonometry and calculus. By understanding the relationship between degrees and radians, we can convert angles from degrees to radians and vice versa. In this article, we focused on converting $-135^{\circ}$ to radians and provided a step-by-step guide on how to do it. We also discussed common mistakes to avoid and real-world applications of converting degrees to radians.

===========================================================

Q: What is the formula to convert degrees to radians?

---

A: The formula to convert degrees to radians is:

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

Q: How do I convert $-135^{\circ}$ to radians?

---

A: To convert $-135^{\circ}$ to radians, we use the formula:

$$\text{radians} = \frac{\pi}{180} \times (-135^{\circ})$$

$$\text{radians} = \frac{\pi}{180} \times (-135)$$

$$\text{radians} = -\frac{135\pi}{180}$$

$$\text{radians} = -\frac{3\pi}{4}$$

Q: What is the difference between degrees and radians?

---

A: The main difference between degrees and radians is the unit of measurement. Degrees are a unit of measurement for angles, while radians are a unit of measurement for angles in terms of the ratio of the arc length to the radius of a circle.

Q: Why do we need to convert degrees to radians?

---

A: We need to convert degrees to radians because many mathematical formulas and calculations are performed in radians. For example, in trigonometry and calculus, angles are often measured 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 to convert degrees to radians.

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:

• Not using the correct formula: Make sure to use the formula $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ to convert degrees to radians.
• Not simplifying the answer: Simplify the answer by dividing both the numerator and the denominator by their greatest common divisor.
• Not checking the units: Make sure to check the units of the answer to ensure that it is in radians.

Q: How do I convert radians to degrees?

---

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

• Navigation: In navigation, angles are often measured in degrees, but calculations are often performed in radians.
• Engineering: In engineering, angles are often measured in degrees, but calculations are often performed in radians.
• Science: In science, angles are often measured in degrees, but calculations are often performed in radians.

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

---

A: Yes, you can use a calculator to convert radians to degrees. Most calculators have a built-in function to convert radians to degrees.

Conclusion

---

In conclusion, converting degrees to radians is a fundamental concept in mathematics, particularly in trigonometry and calculus. By understanding the relationship between degrees and radians, we can convert angles from degrees to radians and vice versa. In this article, we provided a step-by-step guide on how to convert degrees to radians and answered some frequently asked questions.