The Point { P(x, Y)$}$ Lies On The Terminal Side Of An Angle { \theta = -60^{\circ}$}$ In Standard Position. What Are The Signs Of The Values Of { X$}$ And { Y$}$?A. Both { X$}$ And { Y$}$ Are
Introduction
In trigonometry, the point P(x, y) lies on the terminal side of an angle θ in standard position. The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis. The terminal side of an angle is the side that does not contain the initial side. In this article, we will explore the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position.
Understanding the Standard Position
The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis. This means that the angle is measured counterclockwise from the positive x-axis. When the angle is in standard position, the terminal side of the angle is the side that does not contain the initial side.
The Angle θ = -60°
The angle θ = -60° is a negative angle, which means that it is measured clockwise from the positive x-axis. Since the angle is negative, the terminal side of the angle will be in the third quadrant of the coordinate plane.
The Signs of x and y Values
When the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position, the signs of the values of x and y will depend on the quadrant in which the terminal side of the angle lies.
- Quadrant I: In the first quadrant, both x and y values are positive.
- Quadrant II: In the second quadrant, the x value is negative and the y value is positive.
- Quadrant III: In the third quadrant, both x and y values are negative.
- Quadrant IV: In the fourth quadrant, the x value is positive and the y value is negative.
Since the terminal side of the angle θ = -60° lies in the third quadrant, both x and y values will be negative.
Conclusion
In conclusion, when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position, both x and y values will be negative. This is because the terminal side of the angle lies in the third quadrant of the coordinate plane.
Key Takeaways
- The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis.
- The terminal side of an angle is the side that does not contain the initial side.
- When the angle is negative, the terminal side of the angle will be in the third quadrant of the coordinate plane.
- In the third quadrant, both x and y values are negative.
Frequently Asked Questions
Q: What is the standard position of an angle?
A: The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis.
Q: What is the terminal side of an angle?
A: The terminal side of an angle is the side that does not contain the initial side.
Q: What is the sign of x and y values when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position?
A: Both x and y values will be negative.
Q: Why are x and y values negative in the third quadrant?
A: In the third quadrant, both x and y values are negative because the terminal side of the angle lies in this quadrant.
References
Related Articles
- [The Point P(x, y) and the Signs of x and y Values](#the-point-p(x, y) and the signs of x and y values)
- Understanding the Standard Position
- [The Angle θ = -60°](#the-angle-θ = -60°)
- [The Signs of x and y Values](#the-signs-of-x and y values)
The Point P(x, y) and the Signs of x and y Values: Q&A =====================================================
Introduction
In our previous article, we explored the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position. In this article, we will answer some frequently asked questions related to the point P(x, y) and the signs of x and y values.
Q&A
Q: What is the standard position of an angle?
A: The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis.
Q: What is the terminal side of an angle?
A: The terminal side of an angle is the side that does not contain the initial side.
Q: What is the sign of x and y values when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position?
A: Both x and y values will be negative.
Q: Why are x and y values negative in the third quadrant?
A: In the third quadrant, both x and y values are negative because the terminal side of the angle lies in this quadrant.
Q: What is the difference between a positive and negative angle?
A: A positive angle is measured counterclockwise from the positive x-axis, while a negative angle is measured clockwise from the positive x-axis.
Q: How do you determine the quadrant in which the terminal side of an angle lies?
A: To determine the quadrant in which the terminal side of an angle lies, you can use the following rules:
- If the angle is in the first quadrant, both x and y values are positive.
- If the angle is in the second quadrant, the x value is negative and the y value is positive.
- If the angle is in the third quadrant, both x and y values are negative.
- If the angle is in the fourth quadrant, the x value is positive and the y value is negative.
Q: What is the significance of the terminal side of an angle?
A: The terminal side of an angle is significant because it determines the signs of the x and y values of the point P(x, y) that lies on it.
Q: Can you give an example of a point P(x, y) that lies on the terminal side of an angle θ = -60° in standard position?
A: Yes, an example of a point P(x, y) that lies on the terminal side of an angle θ = -60° in standard position is (-3, -4).
Q: How do you find the values of x and y for a given angle and point P(x, y)?
A: To find the values of x and y for a given angle and point P(x, y), you can use the following formulas:
- x = r cos(θ)
- y = r sin(θ) where r is the distance from the origin to the point P(x, y) and θ is the angle in standard position.
Q: What is the relationship between the values of x and y and the angle θ?
A: The values of x and y are related to the angle θ by the following formulas:
- x = r cos(θ)
- y = r sin(θ) where r is the distance from the origin to the point P(x, y) and θ is the angle in standard position.
Conclusion
In conclusion, the point P(x, y) and the signs of x and y values are related to the terminal side of an angle in standard position. By understanding the standard position of an angle, the terminal side of an angle, and the signs of x and y values, you can determine the values of x and y for a given angle and point P(x, y).
Key Takeaways
- The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis.
- The terminal side of an angle is the side that does not contain the initial side.
- The signs of x and y values are determined by the quadrant in which the terminal side of the angle lies.
- The values of x and y are related to the angle θ by the following formulas:
- x = r cos(θ)
- y = r sin(θ) where r is the distance from the origin to the point P(x, y) and θ is the angle in standard position.
Frequently Asked Questions
Q: What is the standard position of an angle?
A: The standard position of an angle is when the vertex of the angle is at the origin (0, 0) and the initial side is along the positive x-axis.
Q: What is the terminal side of an angle?
A: The terminal side of an angle is the side that does not contain the initial side.
Q: What is the sign of x and y values when the point P(x, y) lies on the terminal side of an angle θ = -60° in standard position?
A: Both x and y values will be negative.
Q: Why are x and y values negative in the third quadrant?
A: In the third quadrant, both x and y values are negative because the terminal side of the angle lies in this quadrant.
References
Related Articles
- [The Point P(x, y) and the Signs of x and y Values](#the-point-p(x, y) and the signs of x and y values)
- Understanding the Standard Position
- [The Angle θ = -60°](#the-angle-θ = -60°)
- [The Signs of x and y Values](#the-signs-of-x and y values)