Applying Trigonometry in the Real World: A Practical Approach

Objectives

1. Review the values of sines, cosines, and tangents for angles of 30º, 45º, and 60º.

2. Apply knowledge of trigonometry to calculate the lengths of the sides of right triangles with these angles.

Contextualization

Trigonometry is a fundamental part of mathematics that has various applications in our daily lives. From constructing buildings to developing computer games, understanding sines, cosines, and tangents is essential for solving practical problems in engineering, architecture, and even physics. These concepts enable precise calculations of distances, heights, and angles, facilitating the creation of safe and functional structures. For example, civil engineers use trigonometry to ensure the proper slope of ramps in buildings, while game designers apply these concepts to create realistic movements in three-dimensional environments.

Relevance of the Theme

Understanding trigonometry is crucial in the current context, as its applications are vast and span various professional areas. In civil construction, it ensures the safety and viability of structures; in aviation, it assists in determining flight routes and altitudes; and in video game design, it allows for the creation of realistic graphics. Therefore, mastering these concepts is essential not only for students' academic success but also for their future professional careers.

Basic Trigonometric Values

The values of sines, cosines, and tangents for the angles of 30º, 45º, and 60º are fundamental in trigonometry. These values are frequently used in calculations involving right triangles and have various practical applications in fields like engineering, architecture, and physics.

• Sine of 30º: 1/2

• Cosine of 30º: √3/2

• Tangent of 30º: √3/3

• Sine of 45º: √2/2

• Cosine of 45º: √2/2

• Tangent of 45º: 1

• Sine of 60º: √3/2

• Cosine of 60º: 1/2

• Tangent of 60º: √3

Calculating the Sides of Right Triangles

Using the values of sine, cosine, and tangent, it is possible to calculate the sides of right triangles. This is particularly useful in engineering and architecture problems, where the precision of measurements is crucial.

• The hypotenuse is the side opposite the right angle and is the longest side of the triangle.

• The legs are the sides adjacent to the right angle.

• The sine formula is used to find the length of a leg opposite one of the angles.

• The cosine formula is used to find the length of a leg adjacent to one of the angles.

• The tangent formula is used to find the ratio between the opposite and adjacent legs.

Practical Applications of Trigonometry

Trigonometry is widely applied in various professional areas. In civil construction, aviation, and video game design, trigonometric concepts are used to solve complex problems and ensure the accuracy and safety of operations.

• Civil engineering: calculation of slopes and heights of structures.

• Aviation: determination of flight routes and altitudes.

• Video game design: creation of realistic movements and three-dimensional environments.

Practical Applications

• In civil construction, engineers use trigonometry to calculate the height of buildings and the slope of ramps.
• In aviation, pilots and air traffic controllers apply trigonometry to determine flight routes and altitudes, ensuring the safety of operations.
• In video game design, trigonometry is essential for creating realistic movements and simulating three-dimensional environments.

Key Terms

• Sine: ratio between the opposite leg and the hypotenuse of a right triangle.

• Cosine: ratio between the adjacent leg and the hypotenuse of a right triangle.

• Tangent: ratio between the opposite leg and the adjacent leg of a right triangle.

• Hypotenuse: longest side of a right triangle, opposite the right angle.

• Leg: each of the shorter sides of a right triangle, adjacent to the right angle.

Questions

• How can trigonometry influence the precision and safety of architectural constructions?

• In what ways are the concepts of sine, cosine, and tangent applied in areas outside of engineering and architecture?

• Why is it important to have a solid understanding of basic trigonometric values to solve practical problems in daily life?

Conclusion

To Reflect

Today's lesson showed us the fundamental importance of trigonometry not only for academic success but also in various areas of the labor market. Understanding the values of sine, cosine, and tangent for specific angles (30º, 45º, and 60º) allows us to solve practical problems accurately, whether in constructing buildings, aviation, or video game design. The hands-on activity of building the theodolite was a valuable opportunity to apply these concepts in a real situation, reinforcing the importance of a solid understanding of basic trigonometric lines.

Mini Challenge - Practical Challenge: Measuring the Height of an Object

Use the knowledge acquired about trigonometry to measure the height of a tall object in your house or at school, such as a tree or a building.

• Choose a tall object that you want to measure.
• Position yourself at a known distance from the object.
• Using a protractor, a string with a weight (like a nut), and a straw, create a simple theodolite.
• Look through the straw to align with the top of the object and record the angle formed by the plumb line on the protractor.
• Use the tangent formula (tan(θ) = height/distance) to calculate the height of the object.
• Check your calculations and compare the estimated height with the actual height, if possible.