The Unit Circle and Standard Position

Oct 13, 2025 by windailys.com 49 views

Introduction

In trigonometry, the unit circle is a fundamental concept that plays a crucial role in understanding various mathematical relationships. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. In this article, we will explore the terminal side of an angle in standard position and its intersection with the unit circle. We will use this information to find the value of $\tan (\theta)$ .

The Unit Circle and Standard Position

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The standard position of an angle is when the vertex of the angle is at the origin, and the initial side of the angle lies along the positive x-axis. The terminal side of the angle is the side that extends from the vertex to the point of intersection with the unit circle.

The Terminal Side of an Angle in Standard Position

Given that the terminal side of an angle $\theta$ in standard position intersects the unit circle at $\left(\frac{20}{29}, \frac{21}{29}\right)$ , we can use this information to find the value of $\tan (\theta)$ .

Finding the Value of $\tan (\theta)$

To find the value of $\tan (\theta)$ , we need to use the definition of the tangent function. The tangent function is defined as the ratio of the sine function to the cosine function:

\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}

We can use the coordinates of the point of intersection to find the values of $\sin (\theta)$ and $\cos (\theta)$ .

Finding the Values of $\sin (\theta)$ and $\cos (\theta)$

The coordinates of the point of intersection are $\left(\frac{20}{29}, \frac{21}{29}\right)$ . We can use these coordinates to find the values of $\sin (\theta)$ and $\cos (\theta)$ .

The sine function is defined as the ratio of the y-coordinate to the radius of the unit circle:

\sin (\theta) = \frac{y}{r}

In this case, the y-coordinate is $\frac{21}{29}$ , and the radius of the unit circle is 1. Therefore:

\sin (\theta) = \frac{\frac{21}{29}}{1} = \frac{21}{29}

The cosine function is defined as the ratio of the x-coordinate to the radius of the unit circle:

\cos (\theta) = \frac{x}{r}

In this case, the x-coordinate is $\frac{20}{29}$ , and the radius of the unit circle is 1. Therefore:

\cos (\theta) = \frac{\frac{20}{29}}{1} = \frac{20}{29}

Finding the Value of $\tan (\theta)$

Now that we have found the values of $\sin (\theta)$ and $\cos (\theta)$ , we can use these values to find the value of $\tan (\theta)$ .

The tangent function is defined as the ratio of the sine function to the cosine function:

\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}

Substituting the values of $\sin (\theta)$ and $\cos (\theta)$ , we get:

\tan (\theta) = \frac{\frac{21}{29}}{\frac{20}{29}} = \frac{21}{20}

Therefore, the value of $\tan (\theta)$ is $\frac{21}{20}$ .

Conclusion

In this article, we explored the terminal side of an angle in standard position and its intersection with the unit circle. We used this information to find the value of $\tan (\theta)$ . We found that the value of $\tan (\theta)$ is $\frac{21}{20}$ . This result demonstrates the importance of the unit circle in understanding various mathematical relationships.

References

[1] "Trigonometry" by Michael Corral, 2018.
[2] "Calculus" by Michael Spivak, 2008.

Glossary

Unit circle: A circle with a radius of 1 unit, centered at the origin of a coordinate plane.
Standard position: The position of an angle when the vertex is at the origin, and the initial side lies along the positive x-axis.
Tangent function: The ratio of the sine function to the cosine function.
Sine function: The ratio of the y-coordinate to the radius of the unit circle.
Cosine function: The ratio of the x-coordinate to the radius of the unit circle.
The Terminal Side of an Angle in Standard Position: A Q&A Article ===========================================================

Introduction

In our previous article, we explored the terminal side of an angle in standard position and its intersection with the unit circle. We used this information to find the value of $\tan (\theta)$ . In this article, we will answer some frequently asked questions related to the terminal side of an angle in standard position.

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 that extends from the vertex to the point of intersection with the unit circle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.

Q: How do I find the value of $\tan (\theta)$ ?

A: To find the value of $\tan (\theta)$ , you need to use the definition of the tangent function. The tangent function is defined as the ratio of the sine function to the cosine function:

\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}

You can use the coordinates of the point of intersection to find the values of $\sin (\theta)$ and $\cos (\theta)$ .

Q: What is the sine function?

A: The sine function is the ratio of the y-coordinate to the radius of the unit circle:

\sin (\theta) = \frac{y}{r}

Q: What is the cosine function?

A: The cosine function is the ratio of the x-coordinate to the radius of the unit circle:

\cos (\theta) = \frac{x}{r}

Q: How do I find the values of $\sin (\theta)$ and $\cos (\theta)$ ?

A: To find the values of $\sin (\theta)$ and $\cos (\theta)$ , you need to use the coordinates of the point of intersection. The coordinates of the point of intersection are $\left(\frac{20}{29}, \frac{21}{29}\right)$ . You can use these coordinates to find the values of $\sin (\theta)$ and $\cos (\theta)$ .

Q: What is the value of $\tan (\theta)$ ?

A: The value of $\tan (\theta)$ is $\frac{21}{20}$ .

Q: Why is the unit circle important in understanding various mathematical relationships?

A: The unit circle is important in understanding various mathematical relationships because it provides a way to visualize and relate different mathematical concepts. The unit circle is used to define the sine and cosine functions, which are essential in trigonometry and calculus.

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

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

Navigation: The terminal side of an angle in standard position is used in navigation to determine the direction of a ship or a plane.
Engineering: The terminal side of an angle in standard position is used in engineering to design and analyze mechanical systems.
Physics: The terminal side of an angle in standard position is used in physics to describe the motion of objects.

Conclusion

In this article, we answered some frequently asked questions related to the terminal side of an angle in standard position. We hope that this article has provided you with a better understanding of the terminal side of an angle in standard position and its importance in mathematics and real-world applications.

References

[1] "Trigonometry" by Michael Corral, 2018.
[2] "Calculus" by Michael Spivak, 2008.

Glossary

Terminal side of an angle in standard position: The side that extends from the vertex to the point of intersection with the unit circle.
Unit circle: A circle with a radius of 1 unit, centered at the origin of a coordinate plane.
Sine function: The ratio of the y-coordinate to the radius of the unit circle.
Cosine function: The ratio of the x-coordinate to the radius of the unit circle.
Tangent function: The ratio of the sine function to the cosine function.