Contextualization

Trigonometry, a branch of mathematics that studies the relationships between angles and the measurements of triangles, has surprising and practical applications in our daily lives. From architecture to travel route planning, trigonometric functions are fundamental to various fields of science.

The project at hand aims to review the basic trigonometric lines centered on angles of 30º, 45º, and 60º. By understanding these measurements and their relationships, it is possible to explore the entire universe of trigonometry, essential in high school mathematics and fundamental for subjects like physics.

Introduction

Trigonometry is a field that derives from geometry and is particularly useful when dealing with circles and triangles. At the core of trigonometry are the sine, cosine, and tangent functions. For angles of 30º, 45º, and 60º, these functions have specific values that can be easily memorized, making problem-solving more agile.

These values, which are simple results of the Pythagorean Theorem, are the basis for the study of periodic phenomena that can be modeled from sine and cosine functions. These phenomena range from sound waves to the rotation of planets.

The basic trigonometric lines for angles of 30º, 45º, and 60º are essential for understanding right triangles and circles, which are one of the main means of applying trigonometry. These lines help calculate the lengths of the sides of triangles and better understand physical phenomena such as forces, waves, and geometric optics.

Practical Activity

Activity Title: Trigonometry in Practice: Modeling a Sundial

Project Objective

This project aims to review and consolidate the knowledge of the basic trigonometric lines of 30º, 45º, and 60º through the construction of a sundial. The modeling, assembly, and use of the sundial will allow students to understand in practice the applications of trigonometry.

Detailed Project Description

Each group of students (3 to 5 students) will be responsible for building a sundial, using wood, cardboard, or other material of their choice, with the appropriate measurements for correct time reading. They will take measurements and calculations using trigonometric lines and will explain each step in the final report.

Required Materials

1. Construction material: wood, cardboard, metal, etc.
2. Protractor
3. Ruler
4. Glue or other fixing materials
5. Pencil for markings

Detailed Step-by-Step

1. Research and Planning: Research sundials. How do they work? What measurements are necessary? How does sunlight influence time reading? Plan how your sundial will be.

2. Calculating the Gnomon: Using trigonometric lines, calculate the height that the gnomon (the object that casts the shadow on the sundial) should have. Remember that the tangent of the gnomon's inclination angle should be equal to the value of tan(45º).

3. Marking the Hours: Use trigonometric relationships to mark where each hour will be on the sundial. For this, remember that the position of the sun changes throughout the day (15º per hour) and that, therefore, each hour mark will correspond to a different angle.

4. Sundial Construction: On the chosen base, fix the gnomon at the height calculated in step 2 and mark the hours according to the calculations from step 3. The marking should be done in a way that the gnomon's shadow indicates the correct time.

5. Report: Write a group report containing the introduction, development, conclusion, and bibliography used. Detail each step, justify the decisions made, explain the calculations and trigonometric relationships used.

Project Deliverables

• An exhibition of the sundials produced by each group will be held, where they will explain how they managed to implement theory in practice, and how the sundial can show the correct time during the day.
• Written report with the following topics: Introduction (contextualization of the work and the trigonometry concepts involved), Development (description of the project, including plans, calculations, material choices, and difficulties), Conclusions (what they learned from the project, if the sundial worked and why), and Bibliography.