The Period of the Given Function

Oct 13, 2025 by miodeya.com 114 views

Introduction

In mathematics, periodic functions are a crucial concept in understanding various mathematical models and phenomena. A periodic function is a function that repeats its values at regular intervals, known as the period. The period of a function is the length of the interval after which the function repeats its values. In this article, we will focus on finding the period of a given trigonometric function, specifically the cotangent function.

The Cotangent Function

The cotangent function is defined as the ratio of the cosine function to the sine function. It is denoted by the symbol $\cot x$ and is defined as:

\cot x = \frac{\cos x}{\sin x}

The cotangent function has a period of $\pi$ , which means that the function repeats its values every $\pi$ units.

The Given Function

The given function is:

y=3 \cot (4 x-3 \pi)

This function is a transformation of the cotangent function, where the argument of the cotangent function is $4x-3\pi$ . The coefficient of $x$ is $4$ , and the constant term is $-3\pi$ .

Finding the Period

To find the period of the given function, we need to find the value of $T$ such that:

4x-3\pi \equiv 4x-T\pi \pmod{2\pi}

This means that the argument of the cotangent function, $4x-3\pi$ , should be equivalent to $4x-T\pi$ modulo $2\pi$ . In other words, the two expressions should have the same remainder when divided by $2\pi$ .

Solving for T

To solve for $T$ , we can equate the two expressions:

4x-3\pi \equiv 4x-T\pi \pmod{2\pi}

Subtracting $4x$ from both sides, we get:

-3\pi \equiv -T\pi \pmod{2\pi}

Dividing both sides by $-\pi$ , we get:

3 \equiv T \pmod{2}

Since $T$ must be an integer, we can conclude that:

T = 3

The Period of the Given Function

Now that we have found the value of $T$ , we can conclude that the period of the given function is:

T = \boxed{3} \pi

Therefore, the period of the given function is $3\pi$ .

Conclusion

Q: What is a periodic function?

A: A periodic function is a function that repeats its values at regular intervals, known as the period. The period of a function is the length of the interval after which the function repeats its values.

Q: What is the period of the cotangent function?

A: The period of the cotangent function is $\pi$ . This means that the cotangent function repeats its values every $\pi$ units.

Q: How do you find the period of a given function?

A: To find the period of a given function, you need to find the value of $T$ such that the argument of the function is equivalent to $T\pi$ modulo $2\pi$ . In other words, you need to find the value of $T$ such that the function repeats its values every $T\pi$ units.

Q: What is the difference between the period and the frequency of a function?

A: The period of a function is the length of the interval after which the function repeats its values, while the frequency of a function is the number of times the function repeats its values in a given interval. In other words, the period is the time it takes for the function to repeat its values, while the frequency is the number of repetitions per unit of time.

Q: Can a function have a period of zero?

A: No, a function cannot have a period of zero. The period of a function is a non-zero value that represents the length of the interval after which the function repeats its values.

Q: Can a function have a period of infinity?

A: Yes, a function can have a period of infinity. This means that the function does not repeat its values at all, and its graph is a single, continuous curve.

Q: How do you determine if a function is periodic or not?

A: To determine if a function is periodic or not, you need to check if the function repeats its values at regular intervals. If the function repeats its values at regular intervals, then it is periodic. If the function does not repeat its values at all, then it is not periodic.

Q: What are some examples of periodic functions?

A: Some examples of periodic functions include:

The sine function: $y = \sin x$
The cosine function: $y = \cos x$
The tangent function: $y = \tan x$
The cotangent function: $y = \cot x$
The secant function: $y = \sec x$
The cosecant function: $y = \csc x$

Q: What are some examples of non-periodic functions?

A: Some examples of non-periodic functions include:

The exponential function: $y = e^x$
The logarithmic function: $y = \log x$
The polynomial function: $y = ax^2 + bx + c$
The rational function: $y = \frac{ax+b}{cx+d}$

Q: Why are periodic functions important in mathematics?

A: Periodic functions are important in mathematics because they are used to model many real-world phenomena, such as the motion of objects, the behavior of electrical circuits, and the growth of populations. They are also used in many mathematical models, such as the Fourier series and the Laplace transform.

Q: How do you use periodic functions in real-world applications?

A: Periodic functions are used in many real-world applications, such as:

Modeling the motion of objects, such as the motion of a pendulum or the motion of a spring
Modeling the behavior of electrical circuits, such as the behavior of a resistor, a capacitor, or an inductor
Modeling the growth of populations, such as the growth of a population of bacteria or the growth of a population of animals
Modeling the behavior of physical systems, such as the behavior of a pendulum or the behavior of a spring

Q: What are some common mistakes to avoid when working with periodic functions?

A: Some common mistakes to avoid when working with periodic functions include:

Assuming that a function is periodic when it is not
Assuming that a function is not periodic when it is
Failing to account for the period of a function when using it in a mathematical model
Failing to use the correct period of a function when using it in a real-world application.