Contextualization
Trigonometry is one of the pillars of mathematics. Originating from practical needs such as measuring fields and maritime navigation, this discipline deals with the relationships between angles and the lengths of the sides of a triangle. In particular, we will explore the notions of sine, cosine, and tangent, which are the most basic trigonometric functions.
The sine of an angle in a right triangle is the ratio between the length of the hypotenuse and the side opposite the angle. The cosine is similarly defined as the ratio between the length of the hypotenuse and the side adjacent to the angle. Finally, the tangent is the ratio between the sine and the cosine of the same angle. These concepts are not limited to angles in triangles but are extended to any angle in the plane, opening up a wide range of possibilities and applications.
The concepts of sine, cosine, and tangent are useful not only for solving everyday problems such as calculating the height of a building or the distance of an airplane but are also fundamental in physics, engineering, game programming, and many other fields of science and technology. It is these capabilities that make trigonometry such a fascinating and relevant discipline for us, engaging students worldwide.
Trigonometry is one of those subjects that, although it may seem abstract and disconnected from reality, actually has numerous practical applications. If you have ever used a GPS, played a video game, built a virtual item in a 3D design program, or even listened to digital music, you have used technology that, at some level, incorporates trigonometry concepts. Sound waves, for example, can be mathematically described using sine and cosine functions. Similarly, trigonometry is used in computer graphics for rotation, translation, and scaling of images.
To deepen your knowledge on the subject, we recommend the following resources in Portuguese:
- Khan Academy - Introduction to Trigonometry
- Book: Iezzi, Gelson et al. - Fundamentals of Elementary Mathematics Volume 3: Trigonometry.
- SOS Professor - Lessons on Trigonometry
- YouTube - Me Salva! Basic Trigonometry
Practical Activity
Activity Title: "Creating a Sundial - A Trigonometry Project"
Project Objective
The objective of this project is to build a sundial using basic trigonometry concepts. Through this project, students will have the opportunity to apply the concepts of sine, cosine, and tangent in a practical and fun context.
Detailed Project Description
The groups will build a sundial, also known as a gnomon. To do this, they should research the fundamentals of trigonometry and the concepts of sine, cosine, and tangent. Based on these concepts, they should calculate the angles necessary for the construction of the sundial according to the latitude of the city where the school is located.
Required Materials
- Ruler
- Protractor
- Pencil
- Cardboard or wood
- Permanent marker
- Compass
Detailed Step-by-Step
- Research the latitude of the city where the school is located.
- Based on the latitude, calculate the angle of the gnomon (the part that casts the shadow on the sundial). This can be done using the tangent of the angle, which is equal to the cotangent of the latitude.
- Use the cardboard or wood to create the base of the sundial.
- Using the calculated angle and the protractor, draw the gnomon on the base.
- Mark the lines of the sundial, each representing an hour of the day. Remember that the Sun moves 15 degrees per hour.
- Use the compass to align the sundial with the North.
- Observe the shadow of the gnomon throughout the day and compare it with the local time.
Project Deliverables
After completing the practical activity, students must prepare a report where they should:
Introduction: Explain what trigonometry is, its basic functions sine, cosine, and tangent, and the relationship of these functions with reality. Provide a brief description of the project.
Development: Present the theory used in the construction of the sundial, including the use of trigonometric functions. Detail the entire process of building the sundial, explaining how the trigonometric functions were applied at each stage. After that, describe the results obtained and compare them with the local time.
Conclusion: Reflect on the experience, emphasizing the learning about trigonometric functions and how it applied to the construction of the sundial. Highlight the difficulties encountered, how they were overcome, and what skills were developed throughout the project.
Bibliography: List all sources consulted during the project, including books, websites, videos, among others.
Each group will have one month to develop the sundial and write the report. The report must be clear, complete, organized, and must reflect the students' understanding of the application of trigonometry through practical experimentation.
