Properties of Sine Function

Oct 13, 2025 by windailys.com 93 views

Introduction

In mathematics, the sine function is a fundamental concept in trigonometry, and understanding its range is crucial for solving various problems. The sine function is defined as the ratio of the length of the side opposite to an angle to the length of the hypotenuse in a right-angled triangle. In this article, we will explore the range of the sine function, $y = \sin \theta$ , and discuss the correct answer among the given options.

What is the Range of Sine Function?

The range of a function is the set of all possible output values it can produce for the given input values. In the case of the sine function, $y = \sin \theta$ , the input value is the angle $\theta$ , and the output value is the sine of that angle. The sine function oscillates between positive and negative values, and its range is determined by the maximum and minimum values it can attain.

Properties of Sine Function

The sine function has several important properties that help us understand its range. One of the key properties is that the sine function is periodic, meaning it repeats its values after a certain interval. The period of the sine function is $2\pi$ , which means that the sine function completes one full cycle as the angle $\theta$ ranges from $0$ to $2\pi$ . Another important property of the sine function is that it is an odd function, meaning that $\sin (-\theta) = -\sin \theta$ .

Range of Sine Function

Now that we have discussed the properties of the sine function, let's determine its range. The sine function oscillates between positive and negative values, and its maximum value is $1$ , which occurs when $\theta = \frac{\pi}{2}$ . Similarly, its minimum value is $-1$ , which occurs when $\theta = \frac{3\pi}{2}$ . Since the sine function is periodic, it will repeat these values after every $2\pi$ interval. Therefore, the range of the sine function is all the values between $-1$ and $1$ , inclusive.

Conclusion

In conclusion, the range of the sine function, $y = \sin \theta$ , is all the values between $-1$ and $1$ , inclusive. This is because the sine function oscillates between positive and negative values, and its maximum and minimum values are $1$ and $-1$ , respectively. Therefore, the correct answer among the given options is:

A. [-1, 1]

This answer is supported by the properties of the sine function, including its periodicity and oddness. The sine function is a fundamental concept in trigonometry, and understanding its range is crucial for solving various problems. By analyzing the properties of the sine function, we can determine its range and select the correct answer.

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill

Discussion

What do you think about the range of the sine function? Do you have any questions or concerns about this topic? Share your thoughts and engage in a discussion with others in the comments section below.

Introduction

In our previous article, we discussed the range of the sine function, $y = \sin \theta$ , and concluded that it is all the values between $-1$ and $1$ , inclusive. However, we understand that there may be some questions and concerns about this topic. In this article, we will address some of the frequently asked questions about the range of the sine function and provide answers to help clarify any doubts.

Q1: What is the maximum value of the sine function?

A1: The maximum value of the sine function is $1$ , which occurs when $\theta = \frac{\pi}{2}$ .

Q2: What is the minimum value of the sine function?

A2: The minimum value of the sine function is $-1$ , which occurs when $\theta = \frac{3\pi}{2}$ .

Q3: Why does the sine function oscillate between positive and negative values?

A3: The sine function oscillates between positive and negative values because it is an odd function, meaning that $\sin (-\theta) = -\sin \theta$ . This property causes the sine function to reflect about the x-axis, resulting in oscillations between positive and negative values.

Q4: What is the period of the sine function?

A4: The period of the sine function is $2\pi$ , which means that the sine function completes one full cycle as the angle $\theta$ ranges from $0$ to $2\pi$ .

Q5: Can the sine function attain any value between $-1$ and $1$ ?

A5: Yes, the sine function can attain any value between $-1$ and $1$ . This is because the sine function is continuous and differentiable, and its range is determined by its maximum and minimum values.

Q6: How does the sine function relate to the cosine function?

A6: The sine function and the cosine function are related through the Pythagorean identity, $\sin^2 \theta + \cos^2 \theta = 1$ . This identity shows that the sine function and the cosine function are complementary, and their squares add up to $1$ .

Q7: What are some real-world applications of the sine function?

A7: The sine function has many real-world applications, including:

Modeling periodic phenomena, such as sound waves and light waves
Describing the motion of objects, such as pendulums and springs
Calculating distances and angles in navigation and surveying
Analyzing data in fields such as economics and finance

Conclusion

In conclusion, the range of the sine function, $y = \sin \theta$ , is all the values between $-1$ and $1$ , inclusive. We hope that this Q&A article has helped to clarify any doubts and provide a better understanding of this topic. If you have any further questions or concerns, please feel free to ask.

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Engineers and Scientists" by Donald R. Hill