The Sine and Cosine Functions

Oct 13, 2025 by magicalceylontours.com 120 views

Understanding the Unit Circle and the Tangent Function

The unit circle is a fundamental concept in mathematics, particularly in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle is used to define the trigonometric functions, including sine, cosine, and tangent. In this article, we will explore the unit circle and the tangent function, and determine when the tangent function is undefined.

The Unit Circle and Trigonometric Functions

The unit circle is defined by the equation $x^2 + y^2 = 1$ , where $x$ and $y$ are the coordinates of a point on the circle. The trigonometric functions can be defined in terms of the coordinates of a point on the unit circle. The sine and cosine functions are defined as the ratios of the coordinates of a point on the unit circle to the radius of the circle.

Sine Function: $\sin \theta = \frac{y}{r}$
Cosine Function: $\cos \theta = \frac{x}{r}$

The tangent function is defined as the ratio of the sine and cosine functions.

Tangent Function: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

When is the Tangent Function Undefined?

The tangent function is undefined when the cosine function is equal to zero. This is because the tangent function is a ratio of the sine and cosine functions, and division by zero is undefined.

The Cosine Function is Zero

The cosine function is zero when the point on the unit circle is on the y-axis. This occurs when the angle $\theta$ is equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ .

$\theta = \frac{\pi}{2}$ : The point on the unit circle is at $(0, 1)$ .
$\theta = \frac{3\pi}{2}$ : The point on the unit circle is at $(0, -1)$ .

Conclusion

In conclusion, the tangent function is undefined when the cosine function is equal to zero. This occurs when the point on the unit circle is on the y-axis, which happens when the angle $\theta$ is equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ . Therefore, the correct answer is:

C. $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$

Additional Information

The Sine and Cosine Functions: The sine and cosine functions are defined as the ratios of the coordinates of a point on the unit circle to the radius of the circle.
The Tangent Function: The tangent function is defined as the ratio of the sine and cosine functions.
When is the Tangent Function Undefined?: The tangent function is undefined when the cosine function is equal to zero.

References

"Trigonometry" by Michael Corral: A comprehensive textbook on trigonometry.
"The Unit Circle" by Khan Academy: A video tutorial on the unit circle.
"Trigonometric Functions" by Wolfram MathWorld: A comprehensive article on trigonometric functions.
Frequently Asked Questions: The Unit Circle and the Tangent Function

In this article, we will answer some frequently asked questions about the unit circle and the tangent function.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is used to define the trigonometric functions, including sine, cosine, and tangent.

Q: How is the unit circle related to the trigonometric functions?

A: The trigonometric functions can be defined in terms of the coordinates of a point on the unit circle. The sine and cosine functions are defined as the ratios of the coordinates of a point on the unit circle to the radius of the circle.

Q: When is the tangent function undefined?

A: The tangent function is undefined when the cosine function is equal to zero. This occurs when the point on the unit circle is on the y-axis, which happens when the angle $\theta$ is equal to $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ .

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ . This means that the sine and cosine functions are perpendicular to each other, and their squares add up to 1.

Q: How can I visualize the unit circle?

A: You can visualize the unit circle by plotting the points on the circle for different values of $\theta$ . You can also use a graphing calculator or a computer program to visualize the unit circle.

Q: What are some common mistakes to avoid when working with the unit circle?

A: Some common mistakes to avoid when working with the unit circle include:

Confusing the sine and cosine functions: Make sure to remember that the sine function is the ratio of the y-coordinate to the radius, while the cosine function is the ratio of the x-coordinate to the radius.
Forgetting to use the correct quadrant: Make sure to use the correct quadrant when working with the unit circle. For example, if $\theta$ is in the second quadrant, the sine function will be positive and the cosine function will be negative.
Not using the Pythagorean identity: Make sure to use the Pythagorean identity to relate the sine and cosine functions.

Q: How can I apply the unit circle to real-world problems?

A: The unit circle has many real-world applications, including:

Navigation: The unit circle can be used to calculate distances and angles in navigation.
Physics: The unit circle can be used to describe the motion of objects in physics.
Engineering: The unit circle can be used to design and analyze systems in engineering.

Conclusion

In conclusion, the unit circle and the tangent function are fundamental concepts in mathematics. By understanding the unit circle and the tangent function, you can solve problems in trigonometry and apply the concepts to real-world problems.

Additional Resources

"Trigonometry" by Michael Corral: A comprehensive textbook on trigonometry.
"The Unit Circle" by Khan Academy: A video tutorial on the unit circle.
"Trigonometric Functions" by Wolfram MathWorld: A comprehensive article on trigonometric functions.

References

"Trigonometry" by Michael Corral: A comprehensive textbook on trigonometry.
"The Unit Circle" by Khan Academy: A video tutorial on the unit circle.
"Trigonometric Functions" by Wolfram MathWorld: A comprehensive article on trigonometric functions.