Values of Sine Function for Common Angles

Oct 13, 2025 by miodeya.com 144 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental functions in trigonometry is the sine function, denoted by $\sin(x)$ . In this article, we will explore the sine function and its values for different angles.

The Sine Function

The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function can be represented mathematically as:

\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}

Values of Sine Function for Common Angles

In this section, we will complete the table with the values of the sine function for different angles.

Table of Sine Values

$x$	$-180^{\circ}$	$-135^{\circ}$	$-90^{\circ}$	$-45^{\circ}$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$
$\sin(x)$

Calculating Sine Values

To calculate the sine values for the given angles, we can use the unit circle or the trigonometric identities.

For $x = -180^{\circ}$ , the sine value is 0, since the angle is a multiple of $180^{\circ}$ .
For $x = -135^{\circ}$ , the sine value is $-\frac{\sqrt{6}}{2}$ , since the angle is in the third quadrant and the sine function is negative.
For $x = -90^{\circ}$ , the sine value is 0, since the angle is a multiple of $90^{\circ}$ .
For $x = -45^{\circ}$ , the sine value is $-\frac{\sqrt{2}}{2}$ , since the angle is in the third quadrant and the sine function is negative.
For $x = 0^{\circ}$ , the sine value is 0, since the angle is a multiple of $180^{\circ}$ .
For $x = 45^{\circ}$ , the sine value is $\frac{\sqrt{2}}{2}$ , since the angle is in the first quadrant and the sine function is positive.
For $x = 90^{\circ}$ , the sine value is 1, since the angle is a multiple of $90^{\circ}$ .
For $x = 135^{\circ}$ , the sine value is $\frac{\sqrt{6}}{2}$ , since the angle is in the second quadrant and the sine function is positive.
For $x = 180^{\circ}$ , the sine value is 0, since the angle is a multiple of $180^{\circ}$ .

Completed Table

$x$	$-180^{\circ}$	$-135^{\circ}$	$-90^{\circ}$	$-45^{\circ}$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$
$\sin(x)$	0	$-\frac{\sqrt{6}}{2}$	0	$-\frac{\sqrt{2}}{2}$	0	$\frac{\sqrt{2}}{2}$	1	$\frac{\sqrt{6}}{2}$	0

Conclusion

In this article, we have explored the sine function and its values for different angles. We have completed the table with the sine values for the given angles and calculated the values using the unit circle or trigonometric identities. The sine function is a fundamental function in trigonometry and is used to describe the relationships between the sides and angles of triangles.

Applications of Sine Function

The sine function has numerous applications in various fields, including:

Physics: The sine function is used to describe the motion of objects, such as the oscillations of a pendulum or the vibrations of a string.
Engineering: The sine function is used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
Computer Science: The sine function is used in computer graphics, game development, and other applications that require mathematical modeling.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the sine function.

Q: What is the sine function?

A: The sine function is a mathematical function that describes the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

Q: What are the values of the sine function for common angles?

A: The values of the sine function for common angles are:

$x$	$-180^{\circ}$	$-135^{\circ}$	$-90^{\circ}$	$-45^{\circ}$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$
$\sin(x)$	0	$-\frac{\sqrt{6}}{2}$	0	$-\frac{\sqrt{2}}{2}$	0	$\frac{\sqrt{2}}{2}$	1	$\frac{\sqrt{6}}{2}$	0

Q: How do I calculate the sine value for a given angle?

A: To calculate the sine value for a given angle, you can use the unit circle or trigonometric identities.

Q: What are the applications of the sine function?

A: The sine function has numerous applications in various fields, including:

Physics: The sine function is used to describe the motion of objects, such as the oscillations of a pendulum or the vibrations of a string.
Engineering: The sine function is used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
Computer Science: The sine function is used in computer graphics, game development, and other applications that require mathematical modeling.

Q: What are some common mistakes to avoid when working with the sine function?

A: Some common mistakes to avoid when working with the sine function include:

Confusing the sine function with the cosine function: The sine function and the cosine function are related but distinct functions.
Not considering the quadrant: The sine function has different values in different quadrants.
Not using the correct trigonometric identities: Using the wrong trigonometric identities can lead to incorrect results.

Q: How do I use the sine function in real-world applications?

A: The sine function can be used in a variety of real-world applications, including:

Designing and analyzing electrical circuits: The sine function is used to describe the behavior of electrical circuits.
Modeling the motion of objects: The sine function is used to describe the motion of objects, such as the oscillations of a pendulum or the vibrations of a string.
Creating computer graphics: The sine function is used in computer graphics to create realistic images and animations.

Conclusion

In this article, we have answered some of the most frequently asked questions about the sine function. We have covered topics such as the definition of the sine function, its values for common angles, and its applications in various fields. We hope that this article has been helpful in clarifying any questions you may have had about the sine function.