Finding Values for Negative Angles

Oct 13, 2025 by windailys.com 35 views

Introduction

Trigonometric functions are essential in mathematics, particularly in the study of triangles and waves. These functions describe the relationships between the angles and side lengths of triangles. In this article, we will focus on completing a table that lists the values of trigonometric functions for various angles. This will help us understand the behavior of these functions and their applications in real-world problems.

Trigonometric Functions

Before we proceed, let's briefly review the basic trigonometric functions:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Completing the Table

The table below lists the values of sine, cosine, and tangent for various angles. We will complete the table by finding the values of these functions for the given angles.

$x$	$-180^{\circ}$	$-135^{\circ}$	$-90^{\circ}$	$-45^{\circ}$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$
$\sin x$
$\cos x$
$\tan x$

Finding Values for Negative Angles

To find the values of sine, cosine, and tangent for negative angles, we can use the following properties:

Even/Odd Properties: $\sin (-x) = -\sin x$ and $\cos (-x) = \cos x$ .
Reference Angle: For any angle $x$ , the reference angle is the positive acute angle between the terminal side of $x$ and the $x$ -axis.

Using these properties, we can find the values of sine, cosine, and tangent for negative angles.

$-180^{\circ}$

The reference angle for $-180^{\circ}$ is $180^{\circ}$ . Since sine is an odd function, we have $\sin (-180^{\circ}) = -\sin 180^{\circ} = 0$ . Similarly, cosine is an even function, so we have $\cos (-180^{\circ}) = \cos 180^{\circ} = -1$ . Finally, tangent is an odd function, so we have $\tan (-180^{\circ}) = -\tan 180^{\circ} = 0$ .

$-135^{\circ}$

The reference angle for $-135^{\circ}$ is $45^{\circ}$ . Since sine is an odd function, we have $\sin (-135^{\circ}) = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2}$ . Similarly, cosine is an even function, so we have $\cos (-135^{\circ}) = \cos 45^{\circ} = \frac{\sqrt{2}}{2}$ . Finally, tangent is an odd function, so we have $\tan (-135^{\circ}) = -\tan 45^{\circ} = -1$ .

$-90^{\circ}$

The reference angle for $-90^{\circ}$ is $90^{\circ}$ . Since sine is an odd function, we have $\sin (-90^{\circ}) = -\sin 90^{\circ} = -1$ . Similarly, cosine is an even function, so we have $\cos (-90^{\circ}) = \cos 90^{\circ} = 0$ . Finally, tangent is an odd function, so we have $\tan (-90^{\circ}) = -\tan 90^{\circ} = 0$ .

$-45^{\circ}$

The reference angle for $-45^{\circ}$ is $45^{\circ}$ . Since sine is an odd function, we have $\sin (-45^{\circ}) = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2}$ . Similarly, cosine is an even function, so we have $\cos (-45^{\circ}) = \cos 45^{\circ} = \frac{\sqrt{2}}{2}$ . Finally, tangent is an odd function, so we have $\tan (-45^{\circ}) = -\tan 45^{\circ} = -1$ .

$0^{\circ}$

The reference angle for $0^{\circ}$ is $0^{\circ}$ . Since sine is an odd function, we have $\sin 0^{\circ} = 0$ . Similarly, cosine is an even function, so we have $\cos 0^{\circ} = 1$ . Finally, tangent is an odd function, so we have $\tan 0^{\circ} = 0$ .

$45^{\circ}$

The reference angle for $45^{\circ}$ is $45^{\circ}$ . Since sine is an odd function, we have $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ . Similarly, cosine is an even function, so we have $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$ . Finally, tangent is an odd function, so we have $\tan 45^{\circ} = 1$ .

$90^{\circ}$

The reference angle for $90^{\circ}$ is $90^{\circ}$ . Since sine is an odd function, we have $\sin 90^{\circ} = 1$ . Similarly, cosine is an even function, so we have $\cos 90^{\circ} = 0$ . Finally, tangent is an odd function, so we have $\tan 90^{\circ} = \infty$ .

$135^{\circ}$

The reference angle for $135^{\circ}$ is $45^{\circ}$ . Since sine is an odd function, we have $\sin 135^{\circ} = \frac{\sqrt{2}}{2}$ . Similarly, cosine is an even function, so we have $\cos 135^{\circ} = -\frac{\sqrt{2}}{2}$ . Finally, tangent is an odd function, so we have $\tan 135^{\circ} = -1$ .

$180^{\circ}$

The reference angle for $180^{\circ}$ is $180^{\circ}$ . Since sine is an odd function, we have $\sin 180^{\circ} = 0$ . Similarly, cosine is an even function, so we have $\cos 180^{\circ} = -1$ . Finally, tangent is an odd function, so we have $\tan 180^{\circ} = 0$ .

Completed Table

$x$	$-180^{\circ}$	$-135^{\circ}$	$-90^{\circ}$	$-45^{\circ}$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$
$\sin x$	0	$-\frac{\sqrt{2}}{2}$	-1	$-\frac{\sqrt{2}}{2}$	0	$\frac{\sqrt{2}}{2}$	1	$\frac{\sqrt{2}}{2}$	0
$\cos x$	-1	$\frac{\sqrt{2}}{2}$	0	$\frac{\sqrt{2}}{2}$	1	$\frac{\sqrt{2}}{2}$	0	$-\frac{\sqrt{2}}{2}$	-1
$\tan x$	0	-1	0	-1	0	1	$\infty$	-1	0

Conclusion

Introduction

Trigonometric functions are essential in mathematics, particularly in the study of triangles and waves. These functions describe the relationships between the angles and side lengths of triangles. In this article, we will answer some frequently asked questions about trigonometric functions, providing a deeper understanding of these concepts and their applications.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: What are the properties of even and odd functions?

A: Even functions have the property that $f(-x) = f(x)$ , while odd functions have the property that $f(-x) = -f(x)$ . In the case of trigonometric functions, sine is an odd function, while cosine is an even function.

Q: What is a reference angle?

A: A reference angle is the positive acute angle between the terminal side of an angle and the $x$ -axis. Reference angles are used to find the values of trigonometric functions for negative angles.

Q: How do I find the values of trigonometric functions for negative angles?

A: To find the values of trigonometric functions for negative angles, you can use the properties of even and odd functions, as well as reference angles. For example, if you want to find the value of sine for a negative angle, you can use the property that $\sin (-x) = -\sin x$ .

Q: What is the relationship between sine and cosine?

A: The relationship between sine and cosine is given by the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ . This identity can be used to find the values of sine and cosine for any angle.

Q: What is the relationship between tangent and sine and cosine?

A: The relationship between tangent and sine and cosine is given by the identity: $\tan x = \frac{\sin x}{\cos x}$ . This identity can be used to find the values of tangent for any angle.

Q: How do I use trigonometric functions in real-world problems?

A: Trigonometric functions are used in a wide range of real-world problems, including:

Navigation: Trigonometric functions are used in navigation to find the position and direction of objects.
Physics: Trigonometric functions are used in physics to describe the motion of objects and the behavior of waves.
Engineering: Trigonometric functions are used in engineering to design and analyze systems, such as bridges and buildings.

Q: What are some common applications of trigonometric functions?

A: Some common applications of trigonometric functions include:

Surveying: Trigonometric functions are used in surveying to find the position and direction of objects.
Medical Imaging: Trigonometric functions are used in medical imaging to reconstruct images of the body.
Computer Graphics: Trigonometric functions are used in computer graphics to create 3D models and animations.

Conclusion

In this article, we answered some frequently asked questions about trigonometric functions, providing a deeper understanding of these concepts and their applications. Trigonometric functions are essential in mathematics, particularly in the study of triangles and waves. They have a wide range of applications in real-world problems, including navigation, physics, engineering, surveying, medical imaging, and computer graphics.