Understanding the Terminal Side of an Angle

Oct 13, 2025 by windailys.com 61 views

Introduction

In trigonometry, the terminal side of an angle in standard position is a crucial concept that helps us understand the relationships between the coordinates of a point and the angle formed by the point with the positive x-axis. 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 = 3π/4 in standard position.

Understanding the Terminal Side of an Angle

The terminal side of an angle in standard position is the side of the angle that lies in the counterclockwise direction from the positive x-axis. The angle is measured in radians, and the terminal side is the side that contains the point P(x, y). In this case, the angle θ = 3π/4 is in the second quadrant, which means that the terminal side lies in the second quadrant.

Signs of x and y Values

When the point P(x, y) lies on the terminal side of an angle θ = 3π/4 in standard position, the signs of the values of x and y can be determined by the quadrant in which the terminal side lies. Since the terminal side lies in the second quadrant, we can use the following rules to determine the signs of x and y:

In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive.
In the third quadrant, the x-coordinate is negative, and the y-coordinate is negative.
In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative.

Applying the Rules to the Given Angle

Now that we have the rules for determining the signs of x and y values, we can apply them to the given angle θ = 3π/4. Since the terminal side lies in the second quadrant, we can conclude that:

The x-coordinate x is negative.
The y-coordinate y is positive.

Conclusion

In conclusion, when the point P(x, y) lies on the terminal side of an angle θ = 3π/4 in standard position, the signs of the values of x and y are negative for x and positive for y. This is because the terminal side lies in the second quadrant, where the x-coordinate is negative and the y-coordinate is positive.

Example Problems

Here are some example problems that illustrate the concept of the terminal side of an angle in standard position:

Example 1

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = π/4 in standard position.

Solution:

Since the terminal side lies in the first quadrant, we can conclude that:

The x-coordinate x is positive.
The y-coordinate y is positive.

Example 2

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 5π/4 in standard position.

Solution:

Since the terminal side lies in the third quadrant, we can conclude that:

The x-coordinate x is negative.
The y-coordinate y is negative.

Example 3

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 7π/4 in standard position.

Solution:

Since the terminal side lies in the fourth quadrant, we can conclude that:

The x-coordinate x is positive.
The y-coordinate y is negative.

Practice Problems

Here are some practice problems that you can use to reinforce your understanding of the terminal side of an angle in standard position:

Practice Problem 1

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 2π/3 in standard position.

Practice Problem 2

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 11π/6 in standard position.

Practice Problem 3

Find the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 13π/4 in standard position.

Solutions to Practice Problems

Here are the solutions to the practice problems:

Solution to Practice Problem 1

Since the terminal side lies in the second quadrant, we can conclude that:

The x-coordinate x is negative.
The y-coordinate y is positive.

Solution to Practice Problem 2

Since the terminal side lies in the third quadrant, we can conclude that:

The x-coordinate x is negative.
The y-coordinate y is negative.

Solution to Practice Problem 3

Since the terminal side lies in the fourth quadrant, we can conclude that:

The x-coordinate x is positive.
The y-coordinate y is negative.

Conclusion

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

A: The terminal side of an angle in standard position is the side of the angle that lies in the counterclockwise direction from the positive x-axis.

Q: How do I determine the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle?

A: To determine the signs of the values of x and y, you need to identify the quadrant in which the terminal side lies. Then, you can use the following rules:

In the first quadrant, both x and y are positive.
In the second quadrant, x is negative and y is positive.
In the third quadrant, both x and y are negative.
In the fourth quadrant, x is positive and y is negative.

Q: What is the significance of the angle θ = 3π/4 in standard position?

A: The angle θ = 3π/4 in standard position is a special angle that lies in the second quadrant. This means that the terminal side lies in the second quadrant, where x is negative and y is positive.

Q: Can I use the same rules to determine the signs of x and y for any angle in standard position?

A: Yes, you can use the same rules to determine the signs of x and y for any angle in standard position. The rules are based on the quadrant in which the terminal side lies, and they apply to all angles in standard position.

Q: How do I find the signs of x and y when the point P(x, y) lies on the terminal side of an angle θ = π/4 in standard position?

A: Since the terminal side lies in the first quadrant, both x and y are positive.

Q: How do I find the signs of x and y when the point P(x, y) lies on the terminal side of an angle θ = 5π/4 in standard position?

A: Since the terminal side lies in the third quadrant, both x and y are negative.

Q: Can I use the terminal side of an angle in standard position to solve problems involving trigonometry?

A: Yes, you can use the terminal side of an angle in standard position to solve problems involving trigonometry. The terminal side is a fundamental concept in trigonometry that helps you understand the relationships between the coordinates of a point and the angle formed by the point with the positive x-axis.

Q: What are some common applications of the terminal side of an angle in standard position?

A: The terminal side of an angle in standard position has many applications in trigonometry, including:

Solving problems involving right triangles
Finding the values of trigonometric functions such as sine, cosine, and tangent
Solving problems involving circular functions such as sine, cosine, and tangent
Finding the values of inverse trigonometric functions such as arcsine, arccosine, and arctangent

Q: How do I practice using the terminal side of an angle in standard position to solve problems?

A: You can practice using the terminal side of an angle in standard position to solve problems by:

Working through example problems and exercises
Using online resources and study guides
Practicing with real-world applications and scenarios
Joining a study group or seeking help from a tutor or teacher

Conclusion

In conclusion, the terminal side of an angle in standard position is a fundamental concept in trigonometry that helps you understand the relationships between the coordinates of a point and the angle formed by the point with the positive x-axis. By applying the rules for determining the signs of x and y values, you can determine the signs of the values of x and y when the point P(x, y) lies on the terminal side of an angle θ = 3π/4 in standard position.