Understanding the Sinusoidal Function

Oct 13, 2025 by thesweetdesigns.com 19 views

Solving the Sinusoidal Equation: Finding the Zeros of y = sin(x)

The sinusoidal function, denoted by y = sin(x), is a fundamental concept in mathematics, particularly in trigonometry. It is a periodic function that oscillates between -1 and 1, with a period of 2π. In this article, we will focus on finding the zeros of the sinusoidal function, which are the values of x that make y = 0. We will explore the different formulas that give the zeros of y = sin(x) and determine the correct answer.

Understanding the Sinusoidal Function

The sinusoidal function, y = sin(x), is a periodic function that can be represented by the following equation:

y = sin(x) = (e^(ix) - e^(-ix)) / (2i)

where e is the base of the natural logarithm, i is the imaginary unit, and x is the input variable.

Finding the Zeros of y = sin(x)

To find the zeros of y = sin(x), we need to determine the values of x that make y = 0. This can be done by setting the numerator of the equation equal to zero and solving for x.

e^(ix) - e^(-ix) = 0

Using the properties of exponents, we can rewrite the equation as:

e^(ix) = e^(-ix)

Taking the natural logarithm of both sides, we get:

ix = -ix

Dividing both sides by i, we get:

x = -x

Multiplying both sides by -1, we get:

x = x

This equation is true for all values of x, but we are interested in finding the zeros of y = sin(x). To do this, we need to consider the periodic nature of the sinusoidal function.

Periodic Nature of the Sinusoidal Function

The sinusoidal function has a period of 2π, which means that the function repeats itself every 2π units. This means that if x is a zero of y = sin(x), then x + 2π is also a zero.

Formulas for the Zeros of y = sin(x)

There are several formulas that give the zeros of y = sin(x). Let's examine each of them:

A. $k \pi$ for any positive integer $k$

This formula states that the zeros of y = sin(x) are given by $k \pi$ for any positive integer $k$ . This means that the zeros of the function are at $0, \pi, 2\pi, 3\pi, ...$ .

B. $k \pi$ for any integer $k$

This formula states that the zeros of y = sin(x) are given by $k \pi$ for any integer $k$ . This means that the zeros of the function are at $0, \pi, -\pi, 2\pi, -2\pi, ...$ .

C. $\frac{k \pi}{2}$ for any positive integer $k$

This formula states that the zeros of y = sin(x) are given by $\frac{k \pi}{2}$ for any positive integer $k$ . This means that the zeros of the function are at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi, ...$ .

D. $\frac{k \pi}{2}$ for any integer $k$

This formula states that the zeros of y = sin(x) are given by $\frac{k \pi}{2}$ for any integer $k$ . This means that the zeros of the function are at $0, \frac{\pi}{2}, -\frac{\pi}{2}, \pi, -\pi, \frac{3\pi}{2}, -\frac{3\pi}{2}, ...$ .

In conclusion, the correct formula for the zeros of y = sin(x) is $k \pi$ for any integer $k$ . This means that the zeros of the function are at $0, \pi, -\pi, 2\pi, -2\pi, ...$ . The other formulas, while close, do not accurately describe the zeros of the function.

$The final answer is B. $k \pi$ for any integer $k$.<br/> **Frequently Asked Questions: Zeros of y = sin(x)**$

Q: What is the period of the sinusoidal function y = sin(x)?

A: The period of the sinusoidal function y = sin(x) is 2π. This means that the function repeats itself every 2π units.

Q: What are the zeros of the sinusoidal function y = sin(x)?

A: The zeros of the sinusoidal function y = sin(x) are given by $k \pi$ for any integer $k$ . This means that the zeros of the function are at $0, \pi, -\pi, 2\pi, -2\pi, ...$ .

Q: Why do the zeros of the sinusoidal function y = sin(x) occur at $k \pi$ for any integer $k$ ?

A: The zeros of the sinusoidal function y = sin(x) occur at $k \pi$ for any integer $k$ because the function has a period of 2π. This means that if x is a zero of the function, then x + 2π is also a zero.

Q: Can the zeros of the sinusoidal function y = sin(x) be found using other formulas?

A: Yes, the zeros of the sinusoidal function y = sin(x) can be found using other formulas, such as $\frac{k \pi}{2}$ for any positive integer $k$ . However, these formulas do not accurately describe the zeros of the function.

Q: What is the significance of the zeros of the sinusoidal function y = sin(x)?

A: The zeros of the sinusoidal function y = sin(x) are significant because they represent the points where the function crosses the x-axis. These points are important in many applications, such as physics and engineering, where the sinusoidal function is used to model periodic phenomena.

Q: How can the zeros of the sinusoidal function y = sin(x) be used in real-world applications?

A: The zeros of the sinusoidal function y = sin(x) can be used in many real-world applications, such as:

Modeling periodic phenomena, such as the motion of a pendulum or the vibration of a spring
Analyzing the behavior of electrical circuits, such as filters and oscillators
Designing control systems, such as those used in robotics and aerospace engineering

Q: What are some common mistakes to avoid when finding the zeros of the sinusoidal function y = sin(x)?

A: Some common mistakes to avoid when finding the zeros of the sinusoidal function y = sin(x) include:

Using the wrong formula, such as $\frac{k \pi}{2}$ for any positive integer $k$
Failing to consider the periodic nature of the function
Not accounting for the fact that the function has zeros at $k \pi$ for any integer $k$

Q: How can the zeros of the sinusoidal function y = sin(x) be visualized?

A: The zeros of the sinusoidal function y = sin(x) can be visualized by plotting the function on a graph. The zeros of the function will appear as points on the x-axis where the function crosses the axis.

Q: What are some real-world examples of the sinusoidal function y = sin(x) in action?

A: Some real-world examples of the sinusoidal function y = sin(x) in action include:

The motion of a pendulum, which can be modeled using the sinusoidal function
The vibration of a spring, which can be modeled using the sinusoidal function
The behavior of electrical circuits, such as filters and oscillators, which can be modeled using the sinusoidal function