Trigonometry in Action: Practical Applications of Trigonometric Ratios

Objectives

1. Recognize the main trigonometric ratios: sine, cosine, and tangent of an angle in a right triangle.

2. Apply the trigonometric ratios of 30º, 45º, and 60º to calculate the length of the sides of a right triangle.

Contextualization

Trigonometric ratios, such as sine, cosine, and tangent, are essential for understanding and solving problems involving right triangles. These ratios are widely applied in various fields, such as engineering, architecture, and technology. For example, a civil engineer may use trigonometry to calculate the slope of a ramp or the height of a building, while a programmer may apply it in developing 3D graphics for games. This knowledge is vital for ensuring accuracy and efficiency in technical and scientific projects.

Relevance of the Theme

The study of trigonometric ratios is fundamental in today's context, as they are indispensable tools in various professions and everyday activities. With the growing demand for precision and efficiency in fields such as engineering, architecture, and technology, mastering these mathematical concepts is becoming increasingly crucial. Moreover, understanding trigonometric ratios provides a solid foundation for future studies in mathematics and other scientific disciplines.

Sine

The sine of an angle in a right triangle is defined as the ratio between the length of the opposite side and the length of the hypotenuse. It is one of the main trigonometric ratios and is fundamental for solving problems involving right triangles.

• Sine is represented by the abbreviation 'sin'.

• It is calculated using the formula: sin(θ) = opposite side / hypotenuse.

• The sine of 30º is 1/2, 45º is √2/2, and 60º is √3/2.

Cosine

The cosine of an angle in a right triangle is the ratio between the length of the adjacent side and the length of the hypotenuse. It is a crucial trigonometric ratio for solving various geometric problems.

• Cosine is represented by the abbreviation 'cos'.

• It is calculated using the formula: cos(θ) = adjacent side / hypotenuse.

• The cosine of 30º is √3/2, 45º is √2/2, and 60º is 1/2.

Tangent

The tangent of an angle in a right triangle is the ratio between the length of the opposite side and the length of the adjacent side. Tangent is widely used in calculating slopes and inclines.

• Tangent is represented by the abbreviation 'tan'.

• It is calculated using the formula: tan(θ) = opposite side / adjacent side.

• The tangent of 30º is √3/3, 45º is 1, and 60º is √3.

Practical Applications

• Civil Engineering: Calculate the slope of ramps and the height of buildings using trigonometric ratios.
• Architecture: Determine angles and lengths in structural designs to ensure stability and aesthetics of constructions.
• Information Technology: Develop 3D graphics for games and animations, using trigonometry to calculate angles and distances.

Key Terms

• Sine: Ratio between the length of the opposite side and the hypotenuse.

• Cosine: Ratio between the length of the adjacent side and the hypotenuse.

• Tangent: Ratio between the length of the opposite side and the adjacent side.

Questions

• How can the knowledge of trigonometric ratios be applied in your future career?

• What are the most common difficulties when working with trigonometry and how can they be overcome?

• How can trigonometry be useful in everyday activities outside the work environment?

Conclusion

To Reflect

Throughout this lesson, we explored trigonometric ratios and their practical applications in various fields such as engineering, architecture, and technology. Understanding sine, cosine, and tangent goes beyond theoretical mathematics; these concepts are essential for solving real problems and executing projects with precision. Through building models and solving practical problems, you were able to see how trigonometry is applied in the world around us. We hope this lesson has shown the importance of these skills and how they can be useful in your future careers.

Mini Challenge - Calculating the Height of a Building with Trigonometry

Use the concepts of sine, cosine, and tangent to determine the height of a building based on the shadow cast and the angle of elevation of the sun.

• Go to a location where you can observe the shadow of a building.
• With the help of a protractor, measure the angle of elevation of the sun.
• Measure the length of the shadow of the building using a ruler or tape measure.
• Use the trigonometric ratios to calculate the height of the building.
• Check your calculations using different angles and shadow lengths to ensure accuracy.