Understanding the Coordinates of the Terminal Side

Oct 13, 2025 by pynkocean.com 46 views

The Terminal Side of an Angle in Standard Position: Understanding the Value of tan θ

In trigonometry, the terminal side of an angle in standard position is a fundamental concept that helps us understand various trigonometric functions. The terminal side of an angle is the line segment that extends from the origin to the point on the unit circle that corresponds to the angle. In this article, we will explore the concept of the terminal side of an angle in standard position and how it relates to the value of tan θ.

What is the Terminal Side of an Angle in Standard Position?

The terminal side of an angle in standard position is the line segment that extends from the origin to the point on the unit circle that corresponds to the angle. The unit circle is a circle with a radius of 1 centered at the origin. The terminal side of an angle is the line segment that connects the origin to the point on the unit circle that corresponds to the angle.

Understanding the Coordinates of the Terminal Side

The coordinates of the terminal side of an angle in standard position are given by the ordered pair (x, y). The x-coordinate represents the horizontal distance from the origin to the point on the unit circle, while the y-coordinate represents the vertical distance from the origin to the point on the unit circle.

The Given Point P(-3, -4)

In this problem, we are given the point P(-3, -4) on the terminal side of an angle in standard position. This means that the x-coordinate of the point is -3 and the y-coordinate is -4.

Recalling the Definition of tan θ

The tangent of an angle θ is defined as the ratio of the sine of the angle to the cosine of the angle. Mathematically, this can be expressed as:

tan θ = sin θ / cos θ

Finding the Value of tan θ

To find the value of tan θ, we need to find the values of sin θ and cos θ. We can use the coordinates of the point P(-3, -4) to find these values.

Finding the Value of sin θ

The sine of an angle θ is defined as the ratio of the y-coordinate of the point on the unit circle to the radius of the unit circle. In this case, the y-coordinate is -4 and the radius is 1. Therefore, we can calculate the value of sin θ as follows:

sin θ = y-coordinate / radius = -4 / 1 = -4

Finding the Value of cos θ

The cosine of an angle θ is defined as the ratio of the x-coordinate of the point on the unit circle to the radius of the unit circle. In this case, the x-coordinate is -3 and the radius is 1. Therefore, we can calculate the value of cos θ as follows:

cos θ = x-coordinate / radius = -3 / 1 = -3

Calculating the Value of tan θ

Now that we have found the values of sin θ and cos θ, we can calculate the value of tan θ using the formula:

tan θ = sin θ / cos θ

Substituting the values of sin θ and cos θ, we get:

tan θ = -4 / -3 = 4/3

In this article, we have explored the concept of the terminal side of an angle in standard position and how it relates to the value of tan θ. We have used the coordinates of the point P(-3, -4) to find the values of sin θ and cos θ, and then calculated the value of tan θ using the formula tan θ = sin θ / cos θ. The final answer is:

The final answer is A. tan θ = -\frac{4}{3}
The Terminal Side of an Angle in Standard Position: Q&A

In our previous article, we explored the concept of the terminal side of an angle in standard position and how it relates to the value of tan θ. In this article, we will answer some frequently asked questions related to this topic.

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

A: The terminal side of an angle in standard position is the line segment that extends from the origin to the point on the unit circle that corresponds to the angle.

Q: How do I find the coordinates of the terminal side of an angle in standard position?

A: To find the coordinates of the terminal side of an angle in standard position, you need to know the values of the sine and cosine of the angle. You can use the unit circle to find these values.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin. It is used to find the values of the sine and cosine of an angle.

Q: How do I find the value of tan θ using the coordinates of the terminal side?

A: To find the value of tan θ using the coordinates of the terminal side, you need to use the formula:

tan θ = sin θ / cos θ

You can find the values of sin θ and cos θ using the coordinates of the terminal side.

Q: What is the relationship between the terminal side of an angle in standard position and the value of tan θ?

A: The terminal side of an angle in standard position is related to the value of tan θ through the formula:

tan θ = sin θ / cos θ

The coordinates of the terminal side are used to find the values of sin θ and cos θ, which are then used to calculate the value of tan θ.

Q: Can I use the terminal side of an angle in standard position to find the values of other trigonometric functions?

A: Yes, you can use the terminal side of an angle in standard position to find the values of other trigonometric functions, such as sin θ, cos θ, and sec θ.

Q: How do I determine the quadrant of an angle in standard position?

A: To determine the quadrant of an angle in standard position, you need to know the values of the sine and cosine of the angle. You can use the unit circle to find these values and determine the quadrant.

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

A: The terminal side of an angle in standard position is a fundamental concept in trigonometry that helps us understand various trigonometric functions and their relationships.

In this article, we have answered some frequently asked questions related to the terminal side of an angle in standard position and its relationship to the value of tan θ. We hope that this article has provided you with a better understanding of this important concept in trigonometry.

Q: What is the terminal side of an angle in standard position?
A: The terminal side of an angle in standard position is the line segment that extends from the origin to the point on the unit circle that corresponds to the angle.
Q: How do I find the coordinates of the terminal side of an angle in standard position?
A: To find the coordinates of the terminal side of an angle in standard position, you need to know the values of the sine and cosine of the angle.
Q: What is the relationship between the terminal side of an angle in standard position and the value of tan θ?
A: The terminal side of an angle in standard position is related to the value of tan θ through the formula: tan θ = sin θ / cos θ.

Terminal side: The line segment that extends from the origin to the point on the unit circle that corresponds to the angle.
Unit circle: A circle with a radius of 1 centered at the origin.
Sine: The ratio of the y-coordinate of the point on the unit circle to the radius of the unit circle.
Cosine: The ratio of the x-coordinate of the point on the unit circle to the radius of the unit circle.
Tangent: The ratio of the sine of the angle to the cosine of the angle.