Understanding the Terminal Side of an Angle in Standard Position

Oct 13, 2025 by magicalceylontours.com 128 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 angle, the x-axis, and the y-axis. When the terminal side of an angle passes through a given point, we can use this information to find the value of the sine of the angle. In this article, we will explore how to find the value of sin θ when the terminal side of an angle in standard position passes through the point P(15, -8).

Understanding the Terminal Side of an Angle in Standard Position

The terminal side of an angle in standard position is the side of the angle that lies in the coordinate plane and is formed by the intersection of the angle with the x-axis and the y-axis. The terminal side is the side that is opposite the angle and is formed by the intersection of the angle with the x-axis and the y-axis. In the standard position, the terminal side is always in the first quadrant, and the angle is measured counterclockwise from the positive x-axis.

Finding the Value of Sin θ

To find the value of sin θ, we need to use the definition of the sine function, which is:

sin θ = y/r

where y is the y-coordinate of the point on the terminal side of the angle, and r is the distance from the origin to the point on the terminal side of the angle.

In this case, the point on the terminal side of the angle is P(15, -8), so the y-coordinate is -8. To find the distance from the origin to the point P, we can use the distance formula:

r = √(x^2 + y^2)

where x is the x-coordinate of the point, and y is the y-coordinate of the point.

Plugging in the values, we get:

r = √(15^2 + (-8)^2) = √(225 + 64) = √289 = 17

Now that we have the value of r, we can find the value of sin θ:

sin θ = y/r = -8/17

Conclusion

In this article, we have shown how to find the value of sin θ when the terminal side of an angle in standard position passes through the point P(15, -8). We used the definition of the sine function and the distance formula to find the value of sin θ, which is -8/17. This result is consistent with the options provided, and we can conclude that the correct answer is:

The value of sin θ is -8/17.

Discussion

The terminal side of an angle in standard position is a fundamental concept in trigonometry, and understanding how to find the value of sin θ when the terminal side passes through a given point is essential for solving problems in trigonometry. In this article, we have provided a step-by-step guide on how to find the value of sin θ, and we have shown how to use the definition of the sine function and the distance formula to find the value of sin θ.

Example Problems

Find the value of sin θ when the terminal side of an angle in standard position passes through the point P(3, 4).
Find the value of sin θ when the terminal side of an angle in standard position passes through the point P(-2, 5).

Solutions to Example Problems

To find the value of sin θ when the terminal side of an angle in standard position passes through the point P(3, 4), we can use the definition of the sine function and the distance formula:

r = √(3^2 + 4^2) = √(9 + 16) = √25 = 5

sin θ = y/r = 4/5
To find the value of sin θ when the terminal side of an angle in standard position passes through the point P(-2, 5), we can use the definition of the sine function and the distance formula:

r = √((-2)^2 + 5^2) = √(4 + 25) = √29

sin θ = y/r = 5/√29

Conclusion

In this article, we have shown how to find the value of sin θ when the terminal side of an angle in standard position passes through a given point. We have used the definition of the sine function and the distance formula to find the value of sin θ, and we have provided step-by-step guides on how to solve example problems. We hope that this article has provided a clear understanding of how to find the value of sin θ when the terminal side of an angle in standard position passes through a given point.

Introduction

In our previous article, we explored how to find the value of sin θ when the terminal side of an angle in standard position passes through a given point. In this article, we will answer some frequently asked questions about the terminal side of an angle in standard position and provide additional examples to help solidify your understanding of the concept.

Q&A

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 coordinate plane and is formed by the intersection of the angle with the x-axis and the y-axis.

Q: How do I find the value of sin θ when the terminal side of an angle in standard position passes through a given point?

A: To find the value of sin θ, you can use the definition of the sine function and the distance formula. The distance formula is:

r = √(x^2 + y^2)

where x is the x-coordinate of the point, and y is the y-coordinate of the point. Once you have the value of r, you can find the value of sin θ:

sin θ = y/r

Q: What if the point on the terminal side of the angle has a negative x-coordinate or a negative y-coordinate?

A: If the point on the terminal side of the angle has a negative x-coordinate or a negative y-coordinate, you can still use the definition of the sine function and the distance formula to find the value of sin θ. However, you will need to take the absolute value of the x-coordinate and the y-coordinate when plugging them into the distance formula.

Q: Can I use the terminal side of an angle in standard position to find the value of cos θ?

A: Yes, you can use the terminal side of an angle in standard position to find the value of cos θ. The cosine function is defined as:

cos θ = x/r

where x is the x-coordinate of the point on the terminal side of the angle, and r is the distance from the origin to the point on the terminal side of the angle.

Q: What if the terminal side of the angle in standard position passes through the origin?

A: If the terminal side of the angle in standard position passes through the origin, the value of sin θ is 0, and the value of cos θ is 1.

Additional Examples

Find the value of sin θ when the terminal side of an angle in standard position passes through the point P(6, -3).
Find the value of sin θ when the terminal side of an angle in standard position passes through the point P(-4, 2).
Find the value of cos θ when the terminal side of an angle in standard position passes through the point P(3, 4).

Solutions to Additional Examples

To find the value of sin θ when the terminal side of an angle in standard position passes through the point P(6, -3), we can use the definition of the sine function and the distance formula:

r = √(6^2 + (-3)^2) = √(36 + 9) = √45

sin θ = y/r = -3/√45
To find the value of sin θ when the terminal side of an angle in standard position passes through the point P(-4, 2), we can use the definition of the sine function and the distance formula:

r = √((-4)^2 + 2^2) = √(16 + 4) = √20

sin θ = y/r = 2/√20
To find the value of cos θ when the terminal side of an angle in standard position passes through the point P(3, 4), we can use the definition of the cosine function and the distance formula:

r = √(3^2 + 4^2) = √(9 + 16) = √25 = 5

cos θ = x/r = 3/5

Conclusion

In this article, we have answered some frequently asked questions about the terminal side of an angle in standard position and provided additional examples to help solidify your understanding of the concept. We hope that this article has provided a clear understanding of how to find the value of sin θ and cos θ when the terminal side of an angle in standard position passes through a given point.