Contextualization

Trigonometry is a branch of mathematics that studies the relationships between the lengths of 2 sides of a right triangle (a triangle where one of the angles measures 90º), for different angles. This study, which may seem very specific, has a surprisingly large number of practical applications in various areas, such as physics, engineering, astronomy, and even music.

Trigonometric lines, such as sine, cosine, and tangent, are mathematical functions that express these relationships. Even considering only acute angles, which are smaller than 90º, we already have a multitude of possibilities. However, in many practical situations, we need to deal with an even greater variety of angles.

To facilitate our calculations and understanding, trigonometry generally relates the theory to angles of 30º, 45º, and 60º. However, in some situations, we need to work with angles that are not these, such as 75°, 15°, and 22.5° for example. And that is exactly the goal of this activity: to explore the calculation and application of trigonometric lines of these angles.

And why is this important? Imagine you are an engineer and need to design a ramp for a car. The ideal slope of the ramp is given by the angle it makes with the horizontal plane, and this can be any angle, not just 30º, 45º, or 60º! Or perhaps you are an astronomer and want to calculate the position of a star in the sky. Celestial coordinates are given by angles that can be very different from the angles we are used to using in trigonometry. Therefore, it is essential to expand our knowledge and skills beyond the simplest cases.

Below are some resources that may be useful to deepen knowledge on the subject:

1. A detailed study on trigonometric lines
2. How to calculate the sine, cosine, and tangent of any angle
3. Khanacademy: complete course on trigonometry

Practical Activity: The Trigonometry Behind Angles

Project Objective

Learn to calculate the trigonometric lines of less common angles, such as 75º, 15º, and 22.5º, and understand how these calculations can be applied to real-world problems.

Project Description

This practical project will be carried out in groups of 3 to 5 students and will last for a total of one month. At the end of the activity, the groups must submit a complete report detailing the process used to solve the problems, the application of trigonometric lines, and the conclusions reached.

The activity consists of solving a set of problems involving the determination of the height of an object, the distance between two points, and the slope of a plane, all using the trigonometric lines of angles of 75º, 15º, and 22.5º.

Necessary Materials

• Pencil and paper for notes
• Scientific Calculator
• Ruler and protractor
• Computer with Internet access for research

Step by Step

1. Study of Trigonometric Lines: Start the project by studying trigonometric lines, such as sine, cosine, and tangent, and how to calculate their values for different angles. This step is crucial for understanding and executing the following steps.

2. Calculation of Trigonometric Lines for 75º, 15º, and 22.5º: After familiarizing yourself with the concepts, calculate the sine, cosine, and tangent of these angles. Record your results, they will be important for the rest of the project.

3. Problem Solving: With the values of the trigonometric lines in hand, each group must solve a set of three problems that require the application of these values:

   • Problem 1: The group must calculate the height of a building using the measurement of the building's shadow and the angle of the Sun, assuming the angle is 75º.
   • Problem 2: The group must determine the distance between two boats at sea knowing that they are looking at a lighthouse at different angles - one at an angle of 15º and the other at an angle of 22.5º.
   • Problem 3: Finally, the team must calculate the slope of a ramp given the angle of 15º with the ground.

4. Writing the Report: With the problems solved, it is time to work on the final report.

   • Introduction: The student must contextualize the theme, its relevance and application in the real world as well as the objective of this project.
   • Development: In this section, the student must explain the theory behind the central theme(s) of the project, explain the activity in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: The student must conclude the work by summarizing its main points, explaining the learnings obtained, and the conclusions drawn about the project.
   • Bibliography: The student must indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

In this way, students should conclude with a deeper understanding of trigonometric lines and how they can be applied in real-life situations. They will also have the opportunity to experience the practical resolution of problems, teamwork, and planning of a medium-term project.