The Basics of Trigonometric Ratios

Oct 13, 2025 by miodeya.com 84 views

In mathematics, trigonometric ratios are used to describe the relationships between the angles and side lengths of triangles. These ratios are essential in various mathematical and real-world applications, including physics, engineering, and navigation. In this article, we will explore the function that has the ratio $\frac{24}{25}$ .

The Basics of Trigonometric Ratios

Trigonometric ratios are defined as the ratio of the length of a side of a right triangle to the length of another side. The most common trigonometric ratios 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.

The Given Ratio: $\frac{24}{25}$

The given ratio is $\frac{24}{25}$ . To determine which trigonometric function has this ratio, we need to analyze the possible combinations of side lengths that would result in this ratio.

Analyzing the Options

Let's analyze the options given:

A. $\tan \beta$ : The tangent function is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. If we assume that the side opposite the angle has a length of 24 and the side adjacent to the angle has a length of 25, then the tangent function would be $\tan \beta = \frac{24}{25}$ .

B. $\cos \beta$ : The cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. If we assume that the side adjacent to the angle has a length of 24 and the hypotenuse has a length of 25, then the cosine function would be $\cos \beta = \frac{24}{25}$ .

C. $\sin \theta$ : The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. If we assume that the side opposite the angle has a length of 24 and the hypotenuse has a length of 25, then the sine function would be $\sin \theta = \frac{24}{25}$ .

D. $\cos \theta$ : The cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. If we assume that the side adjacent to the angle has a length of 24 and the hypotenuse has a length of 25, then the cosine function would be $\cos \theta = \frac{24}{25}$ .

Conclusion

Based on the analysis of the options, we can conclude that the function that has the ratio $\frac{24}{25}$ is the tangent function, $\tan \beta$ . This is because the tangent function is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, and the given ratio matches this definition.

Understanding the Significance of Trigonometric Ratios

Trigonometric ratios are essential in various mathematical and real-world applications. They are used to describe the relationships between the angles and side lengths of triangles, and are used in fields such as physics, engineering, and navigation. In this article, we have explored the function that has the ratio $\frac{24}{25}$ , and have concluded that the tangent function, $\tan \beta$ , is the correct answer.

Real-World Applications of Trigonometric Ratios

Trigonometric ratios have numerous real-world applications. Some examples include:

Navigation: Trigonometric ratios are used in navigation to determine the position and direction of objects.
Physics: Trigonometric ratios are used in physics to describe the motion of objects and to calculate forces and energies.
Engineering: Trigonometric ratios are used in engineering to design and build structures such as bridges and buildings.
Computer Science: Trigonometric ratios are used in computer science to develop algorithms and models for various applications.

Conclusion

In conclusion, trigonometric ratios are essential in mathematics and have numerous real-world applications. The function that has the ratio $\frac{24}{25}$ is the tangent function, $\tan \beta$ . Understanding trigonometric ratios is crucial for success in various fields, and is an essential part of mathematical education.

References

"Trigonometry" by Michael Corral. This is a free online textbook that provides a comprehensive introduction to trigonometry.
"Trigonometric Functions" by Wolfram MathWorld. This is a comprehensive online resource that provides information on trigonometric functions and their applications.

Final Thoughts

In our previous article, we explored the function that has the ratio $\frac{24}{25}$ . In this article, we will answer some frequently asked questions about trigonometric ratios.

Q: What are trigonometric ratios?

A: Trigonometric ratios are defined as the ratio of the length of a side of a right triangle to the length of another side. The most common trigonometric ratios 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 most common trigonometric ratios?

A: The most common trigonometric ratios 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: How are trigonometric ratios used in real-world applications?

A: Trigonometric ratios are used in various real-world applications, including:

Navigation: Trigonometric ratios are used in navigation to determine the position and direction of objects.
Physics: Trigonometric ratios are used in physics to describe the motion of objects and to calculate forces and energies.
Engineering: Trigonometric ratios are used in engineering to design and build structures such as bridges and buildings.
Computer Science: Trigonometric ratios are used in computer science to develop algorithms and models for various applications.

Q: What is the difference between sine, cosine, and tangent?

A: The main difference between sine, cosine, and tangent is the ratio of the sides of a right triangle that they represent.

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: How do I calculate trigonometric ratios?

A: To calculate trigonometric ratios, you can use the following formulas:

Sine (sin): $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine (cos): $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent (tan): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
Sum and difference identities: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
Product-to-sum identities: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Q: How do I use trigonometric ratios in problem-solving?

A: To use trigonometric ratios in problem-solving, you can follow these steps:

Identify the problem: Read the problem and identify the trigonometric ratio that is being asked for.
Draw a diagram: Draw a diagram of the problem to help visualize the situation.
Label the sides: Label the sides of the diagram with the given information.
Use the trigonometric ratio: Use the trigonometric ratio to solve for the unknown side or angle.
Check your answer: Check your answer to make sure it is reasonable and makes sense in the context of the problem.

Conclusion

In conclusion, trigonometric ratios are essential in mathematics and have numerous real-world applications. Understanding trigonometric ratios is crucial for success in various fields, and is an essential part of mathematical education. We hope that this article has provided a clear and concise introduction to trigonometric ratios and their applications.