Contextualization

Mathematics, although often appearing abstract, has a multitude of practical and beautiful applications, especially when we talk about trigonometric functions and periodicity. This theme is highly relevant not only to mathematics itself, but to various other areas, including physics, engineering, computer science, and even the arts.

For example, the sine and cosine functions are essential elements to describe and predict periodic phenomena, that is, those that repeat over time. This includes everything from the movement of a clock pendulum to the daily variations of the tides, and even the way we perceive the differences between musical notes.

By understanding the periodicity of these functions, we gain the ability to represent, interpret, and predict a variety of real-world situations. Mathematics becomes a language that allows us to decode the patterns and rhythms of nature and technology.

Introduction

Our goal with this project is to explore and deepen the understanding of trigonometric functions, especially focusing on the concept of periodicity. When we talk about trigonometric functions, we mainly think of the sine, cosine, and tangent functions. We will see how mathematics allows us to model and predict repetitive and cyclical phenomena.

Periodicity is one of the most essential properties of trigonometric functions. It is what allows them to be used to model situations where a behavior repeats at regular intervals. The period of a function is the interval in which the function completes one full cycle and starts to repeat.

For example, in the sine function, the periodicity is 2π, which means that the sine function repeats at intervals of 2π. Similarly, the cosine function also has a periodicity of 2π. The tangent, on the other hand, has a periodicity of π. Throughout the project, we will see in detail how these periodicities are calculated and why they are so important.

Practical Activity: "Tides and Trigonometry"

Project Objective

The objective of this project is to apply the concepts of trigonometric function and periodicity in the analysis and prediction of tides at a specific location during a month. Students will observe, collect data, analyze, and create a mathematical model that can predict the tides based on a trigonometric function.

Detailed Project Description

The groups, each composed of 3 to 5 students, will work together to collect tide data at a specific location for an entire month. They should collect the time and height of high and low tides daily.

Subsequently, the students will analyze the data, identifying patterns and trends that can be modeled by a trigonometric function. With this information, they will build a mathematical model that can predict the future tide heights.

The project will be carried out over 4 weeks, with each week dedicated to a different part of the project. In the first week, students will familiarize themselves with the topic, review trigonometric functions and periodicity, and plan how data collection will be done. In the second and third weeks, data collection will take place. In the fourth week, students will analyze the collected data and build the mathematical model.

Required Materials

For the completion of this project, the following materials will be needed:

1. Internet access to collect data on tides.
2. Software or applications for data collection and analysis.
3. Pencils, paper, and calculators for data calculations and analysis.

Detailed Step-by-Step for Activity Execution

Students will follow the steps below to carry out the activity:

1. Familiarization with the topic (1st week): Study trigonometric functions and periodicity using the suggested resources. Discuss and define the strategy for data collection.

2. Data collection (2nd and 3rd week): Daily collection of data on high and low tides (time and height) at the chosen location. Record all data in a spreadsheet.

3. Data analysis (4th week): Analyze the collected data, identifying patterns and trends. Determine the periodicity of the tides and the trigonometric function that best fits the data.

4. Construction of the mathematical model (4th week): Use the identified patterns to build a mathematical model that describes and predicts the tides at the chosen location using a trigonometric function.

Project Deliverables

In addition to actively participating in the activities, each group of students will be responsible for delivering:

1. Final Report: A report, individually written by each group member, describing the project experience from data collection to the construction of the mathematical model. The report should include:

   • Introduction: Contextualizing the theme and the project's relevance.
   • Development: Explaining in detail the methodology used in data collection and analysis, the theory behind the chosen trigonometric function and periodicity, and the presentation of the constructed mathematical model. Students should also discuss the results obtained.
   • Conclusion: Highlighting the learnings and conclusions drawn from the project.
   • Bibliography: Indicating the sources they relied on to work on the project.

2. Oral Presentation: A presentation to the class about the project, explaining the work done, the results achieved, and the importance of studying trigonometric functions and periodicity. Each group member must participate in the presentation.

Through this project, students will develop technical skills in function analysis and periodicity identification. Additionally, they will also work on socio-emotional skills such as time management, communication, problem-solving, and creative thinking.