Properties of Trigonometric Functions

Oct 13, 2025 by miodeya.com 121 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on finding exact values for sine and cosine functions for a given angle.

Understanding the Problem

The problem requires us to find the exact values of sine and cosine functions for the angle $\theta = \frac{-7 \pi}{4}$ . To solve this problem, we need to use the unit circle and the properties of trigonometric functions.

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry and is used to define the trigonometric functions.

Properties of Trigonometric Functions

The sine and cosine functions are periodic with a period of $2 \pi$ . This means that the values of these functions repeat every $2 \pi$ radians.

Finding Exact Values for Sine and Cosine

To find the exact values of sine and cosine functions for the angle $\theta = \frac{-7 \pi}{4}$ , we need to use the unit circle and the properties of trigonometric functions.

Finding the Exact Value of Sine

The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. In the unit circle, the sine function is equal to the y-coordinate of the point on the circle.

import math
theta = -7 * math.pi / 4
sin_theta = math.sin(theta)
print(sin_theta)

However, we are not allowed to use decimals in this problem. To find the exact value of sine, we need to use the properties of trigonometric functions.

Since $\theta = \frac{-7 \pi}{4}$ , we can write it as $\theta = -2 \pi - \frac{\pi}{4}$ . This means that the point on the unit circle corresponding to $\theta$ is the same as the point corresponding to $\frac{\pi}{4}$ , but reflected across the x-axis.

Using the properties of trigonometric functions, we know that $\sin (-\theta) = -\sin \theta$ . Therefore, we can write:

\sin \left(\frac{-7 \pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right)

Since $\sin \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ , we can write:

\sin \left(\frac{-7 \pi}{4}\right) = -\frac{1}{\sqrt{2}}

Finding the Exact Value of Cosine

The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. In the unit circle, the cosine function is equal to the x-coordinate of the point on the circle.

import math
theta = -7 * math.pi / 4
cos_theta = math.cos(theta)
print(cos_theta)

However, we are not allowed to use decimals in this problem. To find the exact value of cosine, we need to use the properties of trigonometric functions.

Using the properties of trigonometric functions, we know that $\cos (-\theta) = \cos \theta$ . Therefore, we can write:

\cos \left(\frac{-7 \pi}{4}\right) = \cos \left(\frac{\pi}{4}\right)

Since $\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$ , we can write:

\cos \left(\frac{-7 \pi}{4}\right) = \frac{1}{\sqrt{2}}

Conclusion

In this article, we have found the exact values of sine and cosine functions for the angle $\theta = \frac{-7 \pi}{4}$ . We have used the unit circle and the properties of trigonometric functions to derive the exact values.

The exact value of sine is $-\frac{1}{\sqrt{2}}$ and the exact value of cosine is $\frac{1}{\sqrt{2}}$ .

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Calculus" by Michael Spivak, 2008.

Glossary

Unit circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
Sine function: A trigonometric function defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
Cosine function: A trigonometric function defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
Periodic function: A function that repeats every $2 \pi$ radians.
Trigonometric Functions: Q&A =============================

Introduction

In our previous article, we discussed finding exact values for sine and cosine functions for a given angle. In this article, we will answer some frequently asked questions related to trigonometric functions.

Q: What is the difference between sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that are used to describe the relationships between the sides and angles of triangles. However, they are defined differently. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry and is used to define the trigonometric functions.

Q: How do I find the exact value of a trigonometric function?

A: To find the exact value of a trigonometric function, you need to use the unit circle and the properties of trigonometric functions. You can also use the properties of right triangles and the definitions of the trigonometric functions to find the exact value.

Q: What is the periodicity of trigonometric functions?

A: The periodicity of trigonometric functions is $2 \pi$ . This means that the values of these functions repeat every $2 \pi$ radians.

Q: How do I use the properties of trigonometric functions to find the exact value of a function?

A: To use the properties of trigonometric functions to find the exact value of a function, you need to identify the angle and the quadrant in which the angle lies. You can then use the properties of trigonometric functions to find the exact value of the function.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

$\sin^2 \theta + \cos^2 \theta = 1$
$\tan \theta = \frac{\sin \theta}{\cos \theta}$
$\cot \theta = \frac{\cos \theta}{\sin \theta}$

Q: How do I use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you need to identify the trigonometric functions and the identities that can be used to simplify the expression. You can then use the identities to simplify the expression.

Q: What are some common applications of trigonometry?

A: Some common applications of trigonometry include:

Navigation: Trigonometry is used in navigation to calculate distances and directions.
Physics: Trigonometry is used in physics to describe the motion of objects.
Engineering: Trigonometry is used in engineering to design and build structures.
Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we have answered some frequently asked questions related to trigonometric functions. We have discussed the difference between sine and cosine functions, the unit circle, and the periodicity of trigonometric functions. We have also discussed how to use the properties of trigonometric functions to find the exact value of a function and how to use trigonometric identities to simplify expressions.

References

[1] "Trigonometry" by Michael Corral, 2019.
[2] "Calculus" by Michael Spivak, 2008.

Glossary

Unit circle: A circle with a radius of 1 centered at the origin of the coordinate plane.
Sine function: A trigonometric function defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
Cosine function: A trigonometric function defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
Periodic function: A function that repeats every $2 \pi$ radians.
Trigonometric identity: A mathematical statement that relates two or more trigonometric functions.
Navigation: The use of trigonometry to calculate distances and directions.
Physics: The use of trigonometry to describe the motion of objects.
Engineering: The use of trigonometry to design and build structures.
Computer Science: The use of trigonometry to create 3D graphics and animations.