Introduction

Relevance of the Topic

Trigonometry: Sum and Difference of Arcs is a fundamental topic within the Mathematics discipline. Understanding this operation is the next step after mastering trigonometric functions and their respective inverses, as it is related to identifying signs and performing operations between angles. Moreover, it is a relevant theme in the applications of mathematics to other disciplines such as Physics and Engineering.

Contextualization

Within the Mathematics curriculum, Trigonometry is introduced in the 1st year of High School. From this introduction, we acquire knowledge about trigonometric functions and their inverses, as well as trigonometric ratios in the right triangle. In the 2nd year, we deepen this knowledge with the introduction of trigonometric circles and the relationship between trigonometric functions, and also learn to use the calculator for trigonometric calculations.

Now, in the 3rd year, after mastering the fundamental concepts of Trigonometry, we delve into the study of the sum and difference of arcs. This is a natural extension of what we have already learned, as it allows us to further explore the relationships between angles and trigonometric functions. This topic will serve as a starting point for the study of more advanced topics, such as trigonometric identities and the resolution of trigonometric equations and inequalities. Therefore, it is imperative that we have a solid understanding of this concept.

Theoretical Development

Components

• Complementary Arcs: Two angles are called complementary when the sum of their measures is 90 degrees. This is an essential condition in Trigonometry, and it enables a series of relationships that are explored in the study of the Sum and Difference of Arcs.

• Supplementary Arcs: Supplementary arcs, in turn, are two angles whose sum of measures is 180 degrees. As we will see, this property is also fundamental for the development of the Sum and Difference of Arcs.

• Sum Formula: The sum formula of trigonometric arcs allows us to combine the trigonometric functions of two arcs to find the trigonometric function of the resulting arc.

• Difference Formula: Just like in the case of the sum, the difference formula of arcs gives us the trigonometric function of an arc in terms of the trigonometric functions of two other arcs.

• Trigonometric Identities: Trigonometric identities, especially the double-angle identity and the half-angle identity, are essential tools in the study of the Sum and Difference of Arcs.

Key Terms

• Unit Circle: In Trigonometry discussions, we often use a circle of radius 1, called the unit circle, centered at the origin of a Cartesian coordinate system. Angles measured counterclockwise from the positive x-axis along the unit circle are known as standard angles.

• Complex Function Phase Formula: The phase formula of a complex function is a mathematical expression that allows the complex function to be expressed in terms of the trigonometric functions of arcs.

Examples and Cases

• Specific Arc Sum Case: Consider the case where we need to calculate the sine function of the sum of the arcs 45 degrees and 30 degrees. Using the sum formula of arcs, we can rewrite the sine of the sum as the sine of the arc 45 degrees multiplied by the cosine of the arc 30 degrees, plus the cosine of the arc 45 degrees multiplied by the sine of the arc 30 degrees. Solving for the trigonometric functions of the given arcs, we find that the sine of the sum is equal to the square root of 2 divided by 2.

• Specific Arc Difference Case: Now, suppose we need to calculate the cosine function of the difference between 75 degrees and 15 degrees. Using the difference formula of arcs, we can rewrite the cosine of the difference as the cosine of the arc 75 degrees multiplied by the cosine of the arc 15 degrees, plus the sine of the arc 75 degrees multiplied by the sine of the arc 15 degrees. Solving for the trigonometric functions of the given arcs, we find that the cosine of the difference is equal to the square root of 2 divided by 2.

These examples illustrate the power of the sum and difference of arcs formulas in solving more complex trigonometric calculations. Understanding these formulas and their correct application will allow us to advance in the study of mathematics and its related fields.

Detailed Summary

Key Points

• Definition of Complementary and Supplementary Arcs: Understanding the definition and property of complementary and supplementary arcs is essential for the manipulation of arcs in Trigonometry. This paves the way for the Sum and Difference of Arcs, which is a central concept in the discipline.

• Formulation of Sum and Difference of Arcs Formulas: The sum and difference of arcs formulas, which gave rise to this topic, are fundamental and powerful expressions of Trigonometry. They allowed us to associate the trigonometric functions of arcs not only with their respective values, but with functions of other arcs.

• Exploration of Trigonometric Identities: Deepening our understanding of how to use trigonometric identities, especially the double-angle identity, the half-angle identity, along with the sum and difference of arcs formulas, we can manipulate and simplify trigonometric expressions more effectively.

Conclusions

• Mastery of Sum and Difference of Arcs: With the in-depth study of this topic, students acquire the necessary skills to manipulate and solve complex problems involving trigonometric functions, both academically and practically.

• Integration with Previous and Future Topics: The Sum and Difference of Arcs deepen the understanding of trigonometric functions and establish the connection between previously studied topics (trigonometric functions, inverses, trigonometric circles) and upcoming topics (trigonometric identities, resolution of trigonometric equations and inequalities).

Suggested Exercises

1. Exercise Applying the Sum of Arcs Formula: Calculate the cosine function of the sum of the arcs of 40 degrees and 30 degrees. Use the Sum of Arcs Formula to solve the exercise.

2. Exercise Applying the Difference of Arcs Formula: Calculate the tangent function of the difference between the arcs of 80 degrees and 40 degrees. Use the Difference of Arcs Formula to solve the exercise.

3. Problem-Solving Exercise: In a right triangle, one of the acute angles measures 30 degrees. Determine the sine and cosine of the difference between the measure of the right angle and the measure of the acute angle, using the difference of arcs formula.