The Terminal Side Of An Angle Measuring − 9 Π 5 -\frac{9 \pi}{5} − 5 9 Π ​ Radians Lies In Which Quadrant?A. I B. II C. III D. IV

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, divided by the x-axis and the y-axis. Understanding the location of the terminal side of an angle is crucial in solving trigonometric problems and visualizing the relationships between angles and their corresponding trigonometric functions.

Understanding Quadrants

Before we can determine the quadrant in which the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies, we need to understand the basic properties of quadrants. The four quadrants are labeled as I, II, III, and IV, starting from the top-right quadrant and moving counterclockwise.

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

Measuring Angles in Radians

To determine the quadrant in which the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies, we need to understand how angles are measured in radians. A radian is a unit of angle measurement that is defined as the ratio of the arc length to the radius of a circle.

• Full Circle: A full circle has a circumference of $2 \pi r$, where $r$ is the radius of the circle. The angle subtended by a full circle is $2 \pi$ radians.
• Right Angle: A right angle has an angle measure of $\frac{\pi}{2}$ radians.
• Straight Angle: A straight angle has an angle measure of $\pi$ radians.

Determining the Quadrant

Now that we have a basic understanding of quadrants and angle measurement in radians, we can determine the quadrant in which the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies.

• Negative Angle: The angle $-\frac{9 \pi}{5}$ radians is a negative angle, which means it lies in the third or fourth quadrant.
• Reference Angle: To determine the quadrant, we need to find the reference angle, which is the positive acute angle between the terminal side of the angle and the x-axis.
• Reference Angle Calculation: The reference angle can be calculated by taking the absolute value of the angle and subtracting it from $2 \pi$. In this case, the reference angle is $2 \pi - \left|-\frac{9 \pi}{5}\right| = 2 \pi - \frac{9 \pi}{5} = \frac{\pi}{5}$ radians.

Conclusion

Based on the reference angle calculation, we can determine that the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies in the fourth quadrant.

Final Answer

The final answer is D. IV.

Additional Information

• Quadrant Determination: The quadrant in which the terminal side of an angle lies can be determined by finding the reference angle and using the following rules:
  • If the reference angle is acute (less than $\frac{\pi}{2}$ radians), the terminal side lies in the first or second quadrant.
  • If the reference angle is obtuse (greater than $\frac{\pi}{2}$ radians), the terminal side lies in the third or fourth quadrant.
• Trigonometric Functions: The trigonometric functions sine, cosine, and tangent can be used to determine the quadrant in which the terminal side of an angle lies. For example, if the sine of an angle is positive, the terminal side lies in the first or second quadrant.
  

Introduction

In our previous article, we discussed how to determine the quadrant in which the terminal side of an angle lies. We used the example of an angle measuring $-\frac{9 \pi}{5}$ radians and determined that the terminal side lies in the fourth quadrant. In this article, we will provide a Q&A section to help you better understand the concept and answer any questions you may have.

Q&A

Q1: What is the terminal side of an angle?

A1: The terminal side of an angle is the side of the angle that lies in a specific quadrant.

Q2: How do you determine the quadrant in which the terminal side of an angle lies?

A2: To determine the quadrant, you need to find the reference angle, which is the positive acute angle between the terminal side of the angle and the x-axis. You can then use the following rules: + If the reference angle is acute (less than $\frac{\pi}{2}$ radians), the terminal side lies in the first or second quadrant. + If the reference angle is obtuse (greater than $\frac{\pi}{2}$ radians), the terminal side lies in the third or fourth quadrant.

Q3: What is the reference angle?

A3: The reference angle is the positive acute angle between the terminal side of the angle and the x-axis.

Q4: How do you calculate the reference angle?

A4: To calculate the reference angle, you can take the absolute value of the angle and subtract it from $2 \pi$. For example, if the angle is $-\frac{9 \pi}{5}$ radians, the reference angle is $2 \pi - \left|-\frac{9 \pi}{5}\right| = 2 \pi - \frac{9 \pi}{5} = \frac{\pi}{5}$ radians.

Q5: What is the quadrant in which the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies?

A5: Based on the reference angle calculation, the terminal side of the angle $-\frac{9 \pi}{5}$ radians lies in the fourth quadrant.

Q6: How do you determine the quadrant using trigonometric functions?

A6: You can use the following rules to determine the quadrant using trigonometric functions: + If the sine of an angle is positive, the terminal side lies in the first or second quadrant. + If the cosine of an angle is positive, the terminal side lies in the first or fourth quadrant. + If the tangent of an angle is positive, the terminal side lies in the first or third quadrant.

Q7: What is the significance of the quadrant in which the terminal side of an angle lies?

A7: The quadrant in which the terminal side of an angle lies is important in trigonometry because it determines the sign of the trigonometric functions. For example, if the terminal side lies in the first quadrant, the sine, cosine, and tangent functions will all be positive.

Conclusion

In this Q&A article, we have provided answers to common questions about determining the quadrant in which the terminal side of an angle lies. We hope that this article has helped you better understand the concept and answer any questions you may have.

Final Answer

The final answer is D. IV.

Additional Information

• Quadrant Determination: The quadrant in which the terminal side of an angle lies can be determined by finding the reference angle and using the following rules:
  • If the reference angle is acute (less than $\frac{\pi}{2}$ radians), the terminal side lies in the first or second quadrant.
  • If the reference angle is obtuse (greater than $\frac{\pi}{2}$ radians), the terminal side lies in the third or fourth quadrant.
• Trigonometric Functions: The trigonometric functions sine, cosine, and tangent can be used to determine the quadrant in which the terminal side of an angle lies. For example, if the sine of an angle is positive, the terminal side lies in the first or second quadrant.