Contextualization

Each mathematical topic learned has a purpose, and the sum and difference of arcs is no different. There is a vast application of Trigonometry, the branch of mathematics that operates this concept, in our daily lives. From aviation, maritime navigation, engineering, astronomy, and even biology, trigonometry is present.

Considering the key concept of our project, the sum and difference of arcs, we can envision its applicability in the real world broadly. Whether in determining distances on a map, in engineering for calculating forces and torques, in paths traveled by vehicles with varied directions, and even in calculating areas of polygons by segments. This topic is an essential component in various disciplines and professions.

Sum and Difference of Arcs

Trigonometry is a branch of Mathematics that studies the relationships between the sides and angles of triangles. At its most fundamental level, it describes properties of circles and oscillations. In particular, the sum and difference of arcs formula is a powerful trigonometric formula used to simplify the calculation process in various contexts.

Let $α$ and $β$ be two angles. The sum and difference formulas for the sines and cosines of these angles are defined as:

\begin{align*} \sin(α + β) &= \sin α \cos β + \cos α \sin β \ \sin(α - β) &= \sin α \cos β - \cos α \sin β \ \cos(α + β) &= \cos α \cos β - \sin α \sin β \ \cos(α - β) &= \cos α \cos β + \sin α \sin β \ \end{align*}

Practical Activity

Activity Title: Exploring the Sum and Difference of Arcs

Project Objective

The objective is to allow students to explore, understand, and apply the sum and difference of arcs formulas. The practical scenario will help them understand the real-world application of the concept, which can sometimes seem abstract.

Detailed Project Description

This project requires students to apply the concept of sum and difference of arcs in a real-world scenario. Students will create a "Mathematical GPS" for a robot that is programmed to move in different directions but only has basic sine, cosine, and tangent functions.

Each group must assume that the robot can move in any direction but always moves in straight lines and makes fixed angles with the x-axis (east) of the Cartesian plane. Based on this, each team must create a series of "routes" that the robot will have to follow and calculate the final direction the robot will be facing using the sum and difference of arcs formulas.

Necessary Materials

1. Notebook and pen for notes and calculations
2. Recommended books and online resources
3. Computer or smartphone with internet access
4. Spreadsheet or calculator for calculations (optional)

Detailed Step-by-Step for the Activity

1. Review of Concepts: First, each team member should review the sum and difference of arcs formulas using the recommended resources above.

2. Group Discussion: Group members then meet (in person or online) to discuss what they have learned and clarify any doubts or confusing concepts.

3. Route Definition: Now the group should imagine that their robot is standing at the origin of the Cartesian plane looking east. The group should then decide on a series of movements for the robot. These movements will always be in a straight line and for a fixed distance, followed by a rotation at a fixed angle relative to the current direction. These movements can be clockwise or counterclockwise.

4. Calculation of Final Direction: Using the sum and difference of arcs formulas, the group should now calculate the final direction the robot will be facing after following all the routes. Each "rotation" the robot makes is a case of sum or difference of arcs, depending on whether the rotation is clockwise or counterclockwise.

5. Documentation: Now, students should document the entire process in a report, explaining the scenario, the route they chose, the calculations they made, and the final direction they discovered the robot will be facing. They should also include a discussion on how the sum and difference of arcs formulas were useful for completing the project.

6. Review and Submission: After composing the report, students should do a final review to correct any errors and ensure all necessary parts of the report are present.

Project Deliverables

Students are expected to submit the report in text format. This report should contain:

• Introduction: revisiting the theory of sum and difference of arcs and contextualizing its use in the robot situation.
• Development: explaining the chosen route, the step-by-step calculations performed, and the application of the sum and difference of arcs formulas.
• Conclusions: synthesis of what was learned, reflection on the usefulness of the sum and difference of arcs in daily life, and acquisition of socio-emotional skills.
• Bibliography: citing all resources consulted for the project.

This report should be submitted one week after the project launch and will be evaluated according to the evaluation criteria that will be proposed in the third part of the project.