The Terminal Side Of An Angle Measuring 13 Π 12 \frac{13 \pi}{12} 12 13 Π ​ Radians Lies In Which Quadrant? A. I B. II C. III D. IV

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Introduction

In trigonometry, angles are measured in radians, and the terminal side of an angle refers to the side of the angle that lies in a specific quadrant. Quadrants are the four sections of the coordinate plane, labeled I, II, III, and IV, with the x-axis and y-axis as the boundaries. In this article, we will explore the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians and determine in which quadrant it lies.

Understanding Quadrants

Before we proceed, let's briefly review the quadrants and their characteristics.

• Quadrant I: This quadrant lies in the upper right section of the coordinate plane, where both x and y coordinates are positive.
• Quadrant II: This quadrant lies in the upper left section of the coordinate plane, where the x-coordinate is negative and the y-coordinate is positive.
• Quadrant III: This quadrant lies in the lower left section of the coordinate plane, where both x and y coordinates are negative.
• Quadrant IV: This quadrant lies in the lower right section of the coordinate plane, where the x-coordinate is positive and the y-coordinate is negative.

Measuring Angles in Radians

Angles are measured in radians, and a full rotation is equal to $2 \pi$ radians. To determine the quadrant of the terminal side of an angle, we need to find the reference angle, which is the acute angle between the terminal side and the x-axis.

Finding the Reference Angle

To find the reference angle, we can use the following formula:

$$\text{Reference Angle} = \left| \theta - 2 \pi k \right|$$

where $\theta$ is the angle in radians and $k$ is an integer.

In this case, the angle is $\frac{13 \pi}{12}$ radians. To find the reference angle, we can substitute this value into the formula:

$$\text{Reference Angle} = \left| \frac{13 \pi}{12} - 2 \pi k \right|$$

Simplifying the Expression

To simplify the expression, we can multiply both sides by 12 to eliminate the fraction:

$$12 \text{Reference Angle} = \left| 13 \pi - 24 \pi k \right|$$

Finding the Value of $k$

To find the value of $k$, we need to determine which integer value of $k$ will result in a reference angle that lies in the correct quadrant.

Determining the Quadrant

Since the angle is $\frac{13 \pi}{12}$ radians, we can see that it lies in the third quadrant. To confirm this, we can use the following inequality:

$$\frac{3 \pi}{2} < \frac{13 \pi}{12} < 2 \pi$$

This inequality shows that the angle lies between $\frac{3 \pi}{2}$ and $2 \pi$, which means that it lies in the third quadrant.

Conclusion

In conclusion, the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lies in the third quadrant.

Final Answer

The final answer is $\boxed{C}$.

Frequently Asked Questions

• What is the terminal side an angle measuring $\frac{13 \pi}{12}$ radians?
• In which quadrant does the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lie?
• How do we determine the quadrant of the terminal side of an angle?

Answer to Frequently Asked Questions

• The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians is the side of the angle that lies in the third quadrant.
• The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lies in the third quadrant.
• To determine the quadrant of the terminal side of an angle, we can use the following steps:
  1. Find the reference angle using the formula: $\text{Reference Angle} = \left| \theta - 2 \pi k \right|$.
  2. Simplify the expression and find the value of $k$.
  3. Use the inequality to determine the quadrant of the terminal side of the angle.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Mathematics for the Nonmathematician" by Morris Kline, 2013.

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry

Related Articles

• [1] "The Terminal Side of an Angle Measuring $\frac{7 \pi}{12}$ Radians Lies in Which Quadrant?"
• [2] "The Terminal Side of an Angle Measuring $\frac{19 \pi}{12}$ Radians Lies in Which Quadrant?"
  

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Introduction

In our previous article, we explored the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians and determined that it lies in the third quadrant. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q1: What is the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians?

A1: The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians is the side of the angle that lies in the third quadrant.

Q2: In which quadrant does the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lie?

A2: The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lies in the third quadrant.

Q3: How do we determine the quadrant of the terminal side of an angle?

A3: To determine the quadrant of the terminal side of an angle, we can use the following steps:

1. Find the reference angle using the formula: $\text{Reference Angle} = \left| \theta - 2 \pi k \right|$.
2. Simplify the expression and find the value of $k$.
3. Use the inequality to determine the quadrant of the terminal side of the angle.

Q4: What is the reference angle of an angle measuring $\frac{13 \pi}{12}$ radians?

A4: The reference angle of an angle measuring $\frac{13 \pi}{12}$ radians is $\left| \frac{13 \pi}{12} - 2 \pi k \right|$.

Q5: How do we simplify the expression for the reference angle?

A5: To simplify the expression for the reference angle, we can multiply both sides by 12 to eliminate the fraction:

$$12 \text{Reference Angle} = \left| 13 \pi - 24 \pi k \right|$$

Q6: What is the value of $k$ for an angle measuring $\frac{13 \pi}{12}$ radians?

A6: The value of $k$ for an angle measuring $\frac{13 \pi}{12}$ radians is 1.

Q7: How do we determine the quadrant of the terminal side of an angle using the inequality?

A7: To determine the quadrant of the terminal side of an angle using the inequality, we can use the following inequality:

$$\frac{3 \pi}{2} < \frac{13 \pi}{12} < 2 \pi$$

Q8: What is the final answer for the quadrant of the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians?

A8: The final answer for the quadrant of the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians is $\boxed{C}$.

Conclusion

In conclusion, we have answered some frequently asked questions related to the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians. We have determined that the terminal side lies in the third quadrant and have provided the steps to determine the quadrant of the terminal side of an angle.

Final Answer

The final answer is $\boxed{C}$.

Frequently Asked Questions

• What is the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians?
• In which quadrant does the terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lie?
• How do we determine the quadrant of the terminal side of an angle?

Answer to Frequently Asked Questions

• The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians is the side of the angle that lies in the third quadrant.
• The terminal side of an angle measuring $\frac{13 \pi}{12}$ radians lies in the third quadrant.
• To determine the quadrant of the terminal side of an angle, we can use the following steps:
  1. Find the reference angle using the formula: $\text{Reference Angle} = \left| \theta - 2 \pi k \right|$.
  2. Simplify the expression and find the value of $k$.
  3. Use the inequality to determine the quadrant of the terminal side of the angle.

References

• [1] "Trigonometry" by Michael Corral, 2015.
• [2] "Mathematics for the Nonmathematician" by Morris Kline, 2013.

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry

Related Articles

• [1] "The Terminal Side of an Angle Measuring $\frac{7 \pi}{12}$ Radians Lies in Which Quadrant?"
• [2] "The Terminal Side of an Angle Measuring $\frac{19 \pi}{12}$ Radians Lies in Which Quadrant?"