The Period of a Trigonometric Function

Oct 13, 2025 by windailys.com 128 views

Introduction

In mathematics, trigonometric functions are a crucial part of calculus and analysis. These functions are used to describe the relationships between the angles and side lengths of triangles. In this article, we will be discussing the period and vertical asymptotes of a given trigonometric function, $k(x) = 1 + 3 \tan(2x)$ . We will delve into the properties of this function, exploring its behavior and characteristics.

The Period of a Trigonometric Function

The period of a trigonometric function is the distance along the x-axis over which the function repeats itself. In other words, it is the length of the interval over which the function completes one full cycle. For the function $k(x) = 1 + 3 \tan(2x)$ , we need to find the period.

To find the period of $k(x)$ , we need to consider the period of the tangent function, which is $\pi$ . However, the tangent function is periodic with a period of $\pi$ , but in this case, we have a $2x$ inside the tangent function. This means that the period of $k(x)$ will be half of the period of the tangent function, which is $\frac{\pi}{2}$ .

The Vertical Asymptotes of a Trigonometric Function

Vertical asymptotes are the vertical lines that the graph of a function approaches but never touches. In the case of the tangent function, the vertical asymptotes occur at the values of x that make the denominator of the function equal to zero. For the function $k(x) = 1 + 3 \tan(2x)$ , the vertical asymptotes will occur at the values of x that make the denominator of the tangent function equal to zero.

The denominator of the tangent function is $\cos(2x)$ . Therefore, the vertical asymptotes of $k(x)$ will occur at the values of x that make $\cos(2x) = 0$ . This happens when $2x = \frac{\pi}{2} + k\pi$ , where $k$ is an integer.

Solving for x, we get $x = \frac{\pi}{4} + \frac{k\pi}{2}$ . Therefore, the vertical asymptotes of $k(x)$ occur at the values of x given by the equation $x = \frac{\pi}{4} + \frac{k\pi}{2}$ .

Discussion

In this article, we have discussed the period and vertical asymptotes of the trigonometric function $k(x) = 1 + 3 \tan(2x)$ . We have found that the period of $k(x)$ is $\frac{\pi}{2}$ and the vertical asymptotes occur at the values of x given by the equation $x = \frac{\pi}{4} + \frac{k\pi}{2}$ .

The period and vertical asymptotes of a trigonometric function are important characteristics that can help us understand the behavior of the function. By analyzing these characteristics, we can gain a deeper understanding of the function and its properties.

Conclusion

In conclusion, the period and vertical asymptotes of a trigonometric function are crucial characteristics that can help us understand the behavior of the function. By analyzing these characteristics, we can gain a deeper understanding of the function and its properties. In this article, we have discussed the period and vertical asymptotes of the trigonometric function $k(x) = 1 + 3 \tan(2x)$ and found that the period is $\frac{\pi}{2}$ and the vertical asymptotes occur at the values of x given by the equation $x = \frac{\pi}{4} + \frac{k\pi}{2}$ .

References

[1] "Trigonometric Functions" by Wolfram MathWorld
[2] "Period and Vertical Asymptotes of Trigonometric Functions" by Math Open Reference

Glossary

Period: The distance along the x-axis over which a function repeats itself.
Vertical Asymptotes: The vertical lines that the graph of a function approaches but never touches.
Tangent Function: A trigonometric function that is defined as the ratio of the sine and cosine functions.
Cosine Function: A trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.
Q&A: Understanding the Period and Vertical Asymptotes of a Trigonometric Function ====================================================================================

Introduction

In our previous article, we discussed the period and vertical asymptotes of the trigonometric function $k(x) = 1 + 3 \tan(2x)$ . We found that the period of $k(x)$ is $\frac{\pi}{2}$ and the vertical asymptotes occur at the values of x given by the equation $x = \frac{\pi}{4} + \frac{k\pi}{2}$ . In this article, we will answer some frequently asked questions about the period and vertical asymptotes of a trigonometric function.

Q: What is the period of a trigonometric function?

A: The period of a trigonometric function is the distance along the x-axis over which the function repeats itself. In other words, it is the length of the interval over which the function completes one full cycle.

Q: How do I find the period of a trigonometric function?

A: To find the period of a trigonometric function, you need to consider the period of the individual trigonometric functions that make up the function. For example, the period of the tangent function is $\pi$ , but if you have a $2x$ inside the tangent function, the period will be half of the period of the tangent function, which is $\frac{\pi}{2}$ .

Q: What are vertical asymptotes?

A: Vertical asymptotes are the vertical lines that the graph of a function approaches but never touches. In the case of the tangent function, the vertical asymptotes occur at the values of x that make the denominator of the function equal to zero.

Q: How do I find the vertical asymptotes of a trigonometric function?

A: To find the vertical asymptotes of a trigonometric function, you need to find the values of x that make the denominator of the function equal to zero. For example, if you have a function like $k(x) = 1 + 3 \tan(2x)$ , the vertical asymptotes will occur at the values of x that make $\cos(2x) = 0$ . This happens when $2x = \frac{\pi}{2} + k\pi$ , where $k$ is an integer.

Q: Can you give me an example of how to find the period and vertical asymptotes of a trigonometric function?

A: Let's say we have a function like $k(x) = 1 + 3 \tan(2x)$ . To find the period of this function, we need to consider the period of the tangent function, which is $\pi$ . However, we have a $2x$ inside the tangent function, so the period will be half of the period of the tangent function, which is $\frac{\pi}{2}$ .

To find the vertical asymptotes of this function, we need to find the values of x that make $\cos(2x) = 0$ . This happens when $2x = \frac{\pi}{2} + k\pi$ , where $k$ is an integer. Solving for x, we get $x = \frac{\pi}{4} + \frac{k\pi}{2}$ .

Q: What are some common mistakes to avoid when finding the period and vertical asymptotes of a trigonometric function?

A: Some common mistakes to avoid when finding the period and vertical asymptotes of a trigonometric function include:

Not considering the period of the individual trigonometric functions that make up the function.
Not finding the values of x that make the denominator of the function equal to zero.
Not solving for x correctly when finding the vertical asymptotes.

Conclusion

In conclusion, the period and vertical asymptotes of a trigonometric function are crucial characteristics that can help us understand the behavior of the function. By analyzing these characteristics, we can gain a deeper understanding of the function and its properties. We hope that this Q&A article has been helpful in answering some of the frequently asked questions about the period and vertical asymptotes of a trigonometric function.

References

[1] "Trigonometric Functions" by Wolfram MathWorld
[2] "Period and Vertical Asymptotes of Trigonometric Functions" by Math Open Reference

Glossary

Period: The distance along the x-axis over which a function repeats itself.
Vertical Asymptotes: The vertical lines that the graph of a function approaches but never touches.
Tangent Function: A trigonometric function that is defined as the ratio of the sine and cosine functions.
Cosine Function: A trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.