Putting Things in Context

Introduction

Trig Equations are equations that involve trigonometric functions and, in most cases, do not have a single solution, but an infinite set of solutions. Most common Trig Equations include the functions of sine, cosine and tangent. Solving these equations is essential in many fields of knowledge such as Physics, Engineering and Math itself.

Trig Equations come from Trig Equations, however, unlike equations, the inequality signs are used: greater than (>), less than (<), greater than or equal to (>=), less than or equal to (<=). An example of a Trig Equation would be: sin(x) > 1/2.

In order to solve Trig Equations it is required to have knowledge of the trigonometric circle, radians, the Cartesian Coordinate System and the concept of periodic functions. Furthermore, the mastering of such mathematical tools is useful for the solution of complex problems in many areas of Science.

Putting Things into Context

Trigonometry has countless real-life applications, from being the foundation for sea and air navigation, to modeling natural phenomena such as the variation of tides and circadian cycles. More specifically, solving Trig Equations is applied in Engineering and Physics, for instance, in the study of oscillations, waves, electromagnetism, and quantum mechanics.

A practical example of the use of Trig Equations in Civil Engineering is related to the design of suspension bridges. The shape of the bridge can be described with the use of a trigonometric function, and then inequalities can be used to find the limits of the security of the structure, such as the maximum load the bridge can handle.

Therefore, understanding the concepts and techniques of solving Trig Equations is an asset for those who wish to pursue a career in Science, Technology, Engineering, and Math.

Hands-on Activity

Activity Title

"Unveiling the Sea of Trig Equations"

Project Goal

This project aims to use the concepts and techniques for solving Trig Equations by creating a real-world problem and solving it. Students will work in groups of 3 to 5 to define and solve a problem and write a report detailing the process.

Detailed Project Description

Each group must create a real-life scenario that can be modeled and solved using a Trig Equation. The problem can come from any field such as natural sciences, engineering, economics, etc.

Next, the group must solve the Trig Equation that models their problem, describing all the steps needed to solve such equation and discussing what the result means in the problem context.

Materials

1. Notebooks or papers for drawing and solving the equations.
2. Computer with Internet access for research purposes.
3. Text editing software or platform (Word, Google Docs, etc.) to create the report.

Activity Step-by-Step

1. Form groups of 3 to 5 students.
2. Research and select a real-life problem that can be modeled by a Trig Equation.
3. Write a short description of the chosen problem explaining how it can be modeled using one or more Trig Equations.
4. Solve the equation that describes the problem. Write down all the steps.
5. Discuss and interpret the solution of the equation in the problem context.
6. Write a report on the project including the description of the problem, how it was modeled, the step-by-step solution of the Trig Equation, the discussion of the solution, and the conclusions.

Project Deliverables

The project deliverables are:

1. Final report: The document must contain the description of the chosen problem, how it was modeled using one or more Trig Equations, all the steps for solving such equation, the discussion of the solution and the conclusions.

   It must follow a scientific paper format, with Introduction, Development, Conclusions and References sections.

   On the Introduction section the students must put the chosen problem in context, why it is important, and how it can be modeled with a Trig Equation.

   On the Development section the students must explain in detail how they solved the equation, with all the steps of the process and the techniques and concepts used.

   On the Conclusions section the students must discuss the result obtained for the inequality, what it means in the problem context and what they learned during the project.

   Finally, on the References section, the students must list all the sources used for doing the project.

2. Oral presentation: Each group must prepare a 10-15 minutes oral presentation to be given to the class, where they explain the chosen problem, how it was modeled using a Trig Equation, how the Trig Equation was solved, and the conclusions obtained.

The deadline to turn in the report and do the presentation is 1 month after the project starts.