Recalling the Definition of Standard Position

Oct 13, 2025 by pinkandtrash.com 88 views

The Point P(x, y) and the Terminal Side of an Angle

Understanding the Problem

When dealing with angles in standard position, it's essential to understand the relationship between the angle and the coordinates of a point on its terminal side. In this problem, we're given an angle of ${$ \theta = -60^{\circ}$}$ and asked to determine the signs of the values of ${$ x $}$ and ${$ y $}$ for the point ${$ P(x, y) $}$ on its terminal side.

Recalling the Definition of Standard Position

An angle in standard position is defined as an angle with its vertex at the origin (0, 0) of a coordinate plane and its initial side along the positive x-axis. The terminal side of the angle is the side that extends from the vertex to the point where the angle intersects the coordinate plane.

Understanding the Sign of Coordinates

In a coordinate plane, the sign of a coordinate is determined by its position relative to the origin. If a point lies to the right of the origin, its x-coordinate is positive. If a point lies to the left of the origin, its x-coordinate is negative. Similarly, if a point lies above the origin, its y-coordinate is positive. If a point lies below the origin, its y-coordinate is negative.

Determining the Signs of x and y

Given that the angle ${$ \theta = -60^{\circ}$}$ is in standard position, we can determine the signs of the values of ${$ x $}$ and ${$ y $}$ by analyzing the position of the terminal side of the angle.

Since the angle is negative, the terminal side of the angle lies in the third quadrant of the coordinate plane.
In the third quadrant, both the x and y coordinates are negative.

Therefore, the signs of the values of ${$ x $}$ and ${$ y $}$ are both negative.

Conclusion

In conclusion, when dealing with angles in standard position, it's essential to understand the relationship between the angle and the coordinates of a point on its terminal side. By analyzing the position of the terminal side of the angle, we can determine the signs of the values of ${$ x $}$ and ${$ y $}$. In this problem, we found that the signs of the values of ${$ x $}$ and ${$ y $}$ are both negative.

Key Takeaways

An angle in standard position has its vertex at the origin (0, 0) of a coordinate plane and its initial side along the positive x-axis.
The terminal side of an angle in standard position is the side that extends from the vertex to the point where the angle intersects the coordinate plane.
The sign of a coordinate is determined by its position relative to the origin.
In the third quadrant, both the x and y coordinates are negative.

Common Mistakes to Avoid

Failing to understand the definition of standard position and the relationship between the angle and the coordinates of a point on its terminal side.
Not analyzing the position of the terminal side of the angle to determine the signs of the values of ${$ x $}$ and ${$ y $}$.
Assuming that the signs of the values of ${$ x $}$ and ${$ y $}$ are always positive or always negative.

Real-World Applications

Understanding the signs of the values of ${$ x $}$ and ${$ y $}$ is crucial in various fields such as physics, engineering, and computer science.
In physics, the signs of the values of ${$ x $}$ and ${$ y $}$ are used to determine the direction of forces and velocities.
In engineering, the signs of the values of ${$ x $}$ and ${$ y $}$ are used to design and analyze mechanical systems.
In computer science, the signs of the values of ${$ x $}$ and ${$ y $}$ are used in graphics and game development.

Practice Problems

Given an angle of ${$ \theta = 30^{\circ}$}$ in standard position, determine the signs of the values of ${$ x $}$ and ${$ y $}$ for the point ${$ P(x, y) $}$ on its terminal side.
Given an angle of ${$ \theta = -90^{\circ}$}$ in standard position, determine the signs of the values of ${$ x $}$ and ${$ y $}$ for the point ${$ P(x, y) $}$ on its terminal side.

Solutions

For the angle ${$ \theta = 30^{\circ}$}$, the terminal side of the angle lies in the first quadrant. Therefore, both the x and y coordinates are positive.
For the angle ${$ \theta = -90^{\circ}$}$, the terminal side of the angle lies in the fourth quadrant. Therefore, the x coordinate is positive and the y coordinate is negative.
Q&A: Understanding the Point P(x, y) and the Terminal Side of an Angle

Frequently Asked Questions

Q: What is the definition of standard position in a coordinate plane?

A: An angle in standard position is defined as an angle with its vertex at the origin (0, 0) of a coordinate plane and its initial side along the positive x-axis.

Q: How do I determine the signs of the values of x and y for a point on the terminal side of an angle?

A: To determine the signs of the values of x and y, analyze the position of the terminal side of the angle. If the angle is in the first quadrant, both x and y are positive. If the angle is in the second quadrant, x is positive and y is negative. If the angle is in the third quadrant, both x and y are negative. If the angle is in the fourth quadrant, x is negative and y is positive.

Q: What is the relationship between the angle and the coordinates of a point on its terminal side?

A: The angle and the coordinates of a point on its terminal side are related in that the angle determines the position of the point in the coordinate plane. The terminal side of the angle is the side that extends from the vertex to the point where the angle intersects the coordinate plane.

Q: How do I use the signs of the values of x and y in real-world applications?

A: The signs of the values of x and y are used in various fields such as physics, engineering, and computer science. In physics, the signs of the values of x and y are used to determine the direction of forces and velocities. In engineering, the signs of the values of x and y are used to design and analyze mechanical systems. In computer science, the signs of the values of x and y are used in graphics and game development.

Q: What are some common mistakes to avoid when working with angles in standard position?

A: Some common mistakes to avoid include failing to understand the definition of standard position and the relationship between the angle and the coordinates of a point on its terminal side. Not analyzing the position of the terminal side of the angle to determine the signs of the values of x and y is also a common mistake.

Q: How do I practice working with angles in standard position?

A: Practice problems are an excellent way to practice working with angles in standard position. Try determining the signs of the values of x and y for different angles in standard position. You can also try designing and analyzing mechanical systems using the signs of the values of x and y.

Q: What are some real-world applications of understanding the point P(x, y) and the terminal side of an angle?

A: Understanding the point P(x, y) and the terminal side of an angle has numerous real-world applications. In physics, it is used to determine the direction of forces and velocities. In engineering, it is used to design and analyze mechanical systems. In computer science, it is used in graphics and game development.

Q: How do I determine the signs of the values of x and y for a point on the terminal side of an angle with a negative measure?

A: To determine the signs of the values of x and y for a point on the terminal side of an angle with a negative measure, analyze the position of the terminal side of the angle. If the angle is in the third quadrant, both x and y are negative. If the angle is in the fourth quadrant, x is negative and y is positive.

Q: What is the significance of the terminal side of an angle in standard position?

A: The terminal side of an angle in standard position is the side that extends from the vertex to the point where the angle intersects the coordinate plane. It is significant because it determines the position of the point in the coordinate plane.

Q: How do I use the terminal side of an angle in standard position to determine the signs of the values of x and y?

A: To use the terminal side of an angle in standard position to determine the signs of the values of x and y, analyze the position of the terminal side of the angle. If the angle is in the first quadrant, both x and y are positive. If the angle is in the second quadrant, x is positive and y is negative. If the angle is in the third quadrant, both x and y are negative. If the angle is in the fourth quadrant, x is negative and y is positive.

Q: What are some common misconceptions about the point P(x, y) and the terminal side of an angle?

A: Some common misconceptions about the point P(x, y) and the terminal side of an angle include assuming that the signs of the values of x and y are always positive or always negative. Another common misconception is failing to understand the definition of standard position and the relationship between the angle and the coordinates of a point on its terminal side.