Example Questions and Solutions

Oct 13, 2025 by miodeya.com 30 views

**Understanding the Zeros of Sine Function: A Comprehensive Guide** ===========================================================

Introduction

The sine function is a fundamental concept in mathematics, particularly in trigonometry. It is a periodic function that oscillates between -1 and 1, and its graph is a wave-like curve. One of the key aspects of the sine function is its zeros, which are the points where the function intersects the x-axis. In this article, we will explore the formula that gives the zeros of the sine function.

What are the Zeros of Sine Function?

The zeros of the sine function are the values of x for which the function equals zero. In other words, they are the points where the graph of the sine function intersects the x-axis. To find the zeros of the sine function, we need to solve the equation sin(x) = 0.

Which Formula Gives the Zeros of Sine Function?

The correct formula for the zeros of the sine function is:

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

This formula states that the zeros of the sine function occur at multiples of pi, where k is any integer. This means that the zeros of the sine function are located at x = kπ, where k is an integer.

Why is the Formula $k \pi$ Correct?

The formula $k \pi$ is correct because the sine function has a period of 2π, which means that it repeats itself every 2π units. The zeros of the sine function occur at the points where the function intersects the x-axis, which are the multiples of pi. This is because the sine function is an odd function, which means that sin(-x) = -sin(x). Therefore, the zeros of the sine function occur at x = kπ, where k is an integer.

What are the Implications of the Formula $k \pi$ ?

The formula $k \pi$ has several implications for the sine function. Firstly, it means that the zeros of the sine function are located at regular intervals, which are multiples of pi. This makes it easy to find the zeros of the sine function, as we can simply multiply pi by any integer k. Secondly, the formula $k \pi$ implies that the sine function has an infinite number of zeros, as there are an infinite number of integers k.

Example Questions and Solutions

Here are some example questions and solutions to help you understand the formula $k \pi$ :

Question 1: Find the zeros of the sine function for x = 0 to 2π.

Solution: To find the zeros of the sine function for x = 0 to 2π, we need to find the values of x for which sin(x) = 0. Using the formula $k \pi$ , we can find the zeros of the sine function at x = 0, π, and 2π.

Question 2: Find the zeros of the sine function for x = -2π to 2π.

Solution: To find the zeros of the sine function for x = -2π to 2π, we need to find the values of x for which sin(x) = 0. Using the formula $k \pi$ , we can find the zeros of the sine function at x = -2π, -π, 0, π, and 2π.

Question 3: Find the zeros of the sine function for x = 3π to 5π.

Solution: To find the zeros of the sine function for x = 3π to 5π, we need to find the values of x for which sin(x) = 0. Using the formula $k \pi$ , we can find the zeros of the sine function at x = 3π and 4π.

Conclusion

In conclusion, the formula $k \pi$ gives the zeros of the sine function, where k is any integer. This formula states that the zeros of the sine function occur at multiples of pi, which are the points where the function intersects the x-axis. Understanding this formula is essential for solving problems involving the sine function, and it has several implications for the function's behavior.

Frequently Asked Questions

Here are some frequently asked questions and answers to help you understand the formula $k \pi$ :

Q: What is the period of the sine function?

A: The period of the sine function is 2π.

Q: What are the zeros of the sine function?

A: The zeros of the sine function are the values of x for which sin(x) = 0.

Q: How do I find the zeros of the sine function?

A: To find the zeros of the sine function, you can use the formula $k \pi$ , where k is any integer.

Q: What are the implications of the formula $k \pi$ ?

A: The formula $k \pi$ implies that the zeros of the sine function are located at regular intervals, which are multiples of pi. This makes it easy to find the zeros of the sine function, as we can simply multiply pi by any integer k.

Glossary of Terms

Here are some key terms related to the sine function and its zeros:

Sine function: A periodic function that oscillates between -1 and 1.
Zeros of the sine function: The values of x for which sin(x) = 0.
Period of the sine function: The distance between two consecutive zeros of the sine function.
Formula $k \pi$ : A formula that gives the zeros of the sine function, where k is any integer.

References

Here are some references that you can use to learn more about the sine function and its zeros:

Trigonometry: A branch of mathematics that deals with the relationships between the sides and angles of triangles.
Sine function: A periodic function that oscillates between -1 and 1.
Zeros of the sine function: The values of x for which sin(x) = 0.
Formula $k \pi$ : A formula that gives the zeros of the sine function, where k is any integer.