Context

Theoretical Introduction

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is extremely useful in a variety of fields, from engineering to music, and even in space exploration. In our studies, we will focus on three fundamental trigonometric functions: sine, cosine, and tangent.

The first function, the sine of an angle, is the ratio of the side opposite the angle to the hypotenuse in a right triangle. The second function, the cosine, is the ratio of the side adjacent to the angle to the hypotenuse. The tangent of an angle is the ratio of the sine to the cosine of that angle, or between the opposite side and the adjacent side.

Trigonometric functions have very interesting properties and have a large number of practical applications, from determining the height of a building to predicting ocean tides. Specifically, we will focus on the angles of 30º, 45º, and 60º, which have easily memorable values and are frequently used in practical problems.

Context

Trigonometry is a powerful tool that is used in many different disciplines and professions. Engineers use trigonometry to calculate forces and angles. Architects use it to design buildings and bridges. Astronomers use trigonometry to calculate distances in space and map the universe.

In our daily lives, trigonometry can be used to calculate distances that we cannot measure directly. For example, we can use trigonometry to calculate the height of a mountain, the distance between two cities on a map, or the height of a building.

In addition to its direct application in the exact sciences, Trigonometry is also an excellent way to develop logical thinking and problem-solving skills, which are valuable skills in any discipline or career.

Suggested Resources: To delve deeper into the subject, students can consult the following sources:

1. Book: "Fundamentals of Elementary Mathematics, Volume 1: Sets, Functions", by Gelson Iezzi. This book has clear and objective language, with many solved and proposed exercises.
2. Website: Khan Academy - Mathematics, Trigonometry.
3. Website: Brasil Escola - Mathematics, Trigonometry.
4. YouTube: Matemática Rio with Prof. Rafael Procopio

Practical Activity

Activity Title: Measuring the World with Trigonometry

Project Objective

The objective of this project is to explore and apply the concepts of sine, cosine, and tangent of the angles 30º, 45º, and 60º to solve practical real-world problems. By combining Trigonometry with Geography, the aim is for students to understand the applicability of Mathematics in other disciplines and develop their problem-solving skills, creativity, time management, and communication skills.

Detailed Project Description

Students, in groups of 3 to 5, will be challenged to calculate real distances and heights of geographical points or structures using Trigonometry. They should research and choose three famous locations or constructions (Ex: Statue of Liberty, Mount Everest, Eiffel Tower, etc.) and estimate their heights or distances between two points using only Trigonometry and data accessible online.

Materials Needed

• Internet access for research.
• Calculator.
• Geolocation and mapping software (Google Earth, Google Maps, etc.).

Step by Step

1. The groups must choose three locations/constructions for the project.
2. After choosing, the students should research the data necessary for their calculations, such as the straight-line distance between two geographical points (if they intend to calculate a distance) or the distance from the observer to the point/base of the construction (if they intend to calculate a height).
3. The students must then apply Trigonometry to estimate the desired height or distance, using the sine, cosine, and tangent of the angles 30º, 45º, and 60º as appropriate. Important: Students should document the entire process, including the data collected, the calculations performed, and the formulas applied.
4. After performing the calculations, the students should compare their estimates with the official data (available from reliable sources on the Internet) and reflect on discrepancies, considering possible sources of error and limitations of their approach.
5. The students should then prepare a 15-minute presentation explaining the process they used, the difficulties faced, the results obtained, and an analysis of their work.

Project Delivery

• Presentation: Students should give a presentation of their results to the class. The presentation should include an explanation of the process used, the challenges faced, and how these challenges were overcome. Students should also highlight what they learned from the activity.
• Written Document: In addition to the oral presentation, students should submit a report documenting the entire process. The report should be divided into four parts: Introduction (contextualizing the problem and objectives), Development (describing in detail the methodology applied, the calculations performed, and the results obtained), Conclusions (analyzing the results and reflecting on the lessons learned) and Bibliography (references of the sources of information used).

At the end of the project, students are expected to have a solid understanding of trigonometric functions (especially sine, cosine, and tangent), the usefulness and application of Trigonometry in the real world, and have developed important skills, such as the ability to solve problems, work in teams, communicate effectively, and manage time.