Contextualization

Trigonometry is an area of study in mathematics that deals with the relationships between the sides and angles of triangles. In our daily lives, we use trigonometry to calculate unknown distances and for a better understanding of the world around us. From mapping oceans and forests to building houses, trigonometry plays a fundamental role in our society.

Trigonometric lines are extremely important for understanding Trigonometry. Especially the lines of 75°, 15°, and 22.5°, which, being notable angles, have specific values that play crucial roles in various practical applications, such as architectural projects, engineering, physics, astronomy, and many other areas of science.

Theoretical Introduction

The basis of all trigonometry lies in the sine, cosine, and tangent functions. Specifically, for an angle θ in a right triangle, the sine is the ratio between the length of the opposite side and the hypotenuse. The cosine is the ratio between the adjacent side and the hypotenuse. And the tangent is the ratio between the sine and the cosine.

In the case of the trigonometric lines of 75°, 15°, and 22.5°, these are examples of notable angles. Notable angles are angles whose measures repeat in regular geometric figures and, therefore, have tabulated values for sine, cosine, and tangent. This knowledge allows us to solve problems involving these angles in a faster and more efficient way.

It is important to understand that trigonometry, although a pure field of mathematics, has applications in various disciplines. For example, in physics, trigonometry is used to study vectors, wave phenomena, and rotational mechanics. In engineering and architecture, these relationships are used to design and build structures and machines.

Practical Activity: The Mathematics Behind Art with Trigonometric Lines

Project Objective

The objective of this project is to create a work of art using the mathematics of the trigonometric lines of 75º, 15º, and 22.5º. Students should research and develop practical knowledge in Trigonometry, particularly regarding trigonometric lines, and apply this knowledge in a creative and artistic way. Additionally, this project also aims to develop teamwork skills, time management, and critical thinking.

Detailed Project Description

Students will be divided into groups of 3 to 5 people, and each group will be tasked with creating a work of art that uses the trigonometric lines of 75º, 15º, 22.5º in a fundamental way. The artworks can be anything from a hand-drawn sketch, a painting to a 3D sculpture made with recycled materials. However, the artwork must:

1. Clearly and evidently incorporate the trigonometric lines of 75º, 15º, and 22.5º.
2. Be accompanied by a detailed mathematical explanation describing how the trigonometric lines were used.

Necessary Materials

1. Paper, pens, pencils, and rulers.
2. Materials for creating the artwork (cardboard, paints, brushes, colored pens, clips, glue, tape, recycled materials, etc.)
3. Calculator.
4. Textbooks and internet access for research.

Detailed Step-by-Step

1. Start by reviewing the theory of the trigonometric lines of 75º, 15º, 22.5º and how they are applied. Use the recommended resources and other materials to deepen your understanding.

2. As a group, brainstorm ideas for your art project that incorporate the trigonometric lines of 75º, 15º, 22.5º in a fundamental way.

3. After deciding on an idea, make a detailed sketch of the project. Decide which materials will be needed and how you will obtain them.

4. Create your artwork, making sure to incorporate the trigonometric lines of 75º, 15º, 22.5º clearly and evidently.

5. Alongside the creation of the art, provide a detailed mathematical explanation describing how the trigonometric lines were used in the artwork.

6. Finalize the art and the mathematical explanation, and prepare to present both to the class.

Project Deliverables

At the end of the project, the groups must deliver:

1. The complete artwork, with the trigonometric lines of 75º, 15º, and 22.5º clearly and evidently incorporated.

2. A written report detailing the art creation process, the application of the trigonometric lines, and the meaning behind the artwork. This report should follow the format of Introduction, Development, Conclusions, and Bibliography. When writing this report, students should keep in mind that it complements the practical work done, being a fundamental part of the project. Therefore, it should not be treated as secondary, but as an essential piece that explains in detail the mathematics behind the created art.

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