Introduction

Relevance of the Topic

The trigonometric function is one of the fundamental pillars of mathematics. It describes the relationship between angles and sides of a triangle, which is essential in many fields of study, such as physics, engineering, computer science, among others.

Understanding these functions and their characteristics, including domain, range, and periodicity, is crucial for solving complex problems and interpreting natural phenomena. Thus, exploring the inputs and outputs of these functions, i.e., the values we can input and the results we can obtain, is a central point in mathematics.

Contextualization

The study of the trigonometric function is located in the Mathematics discipline of the 3rd year of High School and is included in the Trigonometry unit. In this unit, fundamental concepts of trigonometry, such as angle measurements, triangles, the relationship between angles and arcs, and of course, trigonometric functions are explored.

After this introductory stage, the focus turns to a detailed analysis of trigonometric functions, their properties, and transformations. The study of inputs and outputs, which is the main point of this lecture note, is a fundamental step for understanding the behavior of these functions and for solving trigonometric equations and inequalities. This is where mathematics begins to show its true power, being applicable to the real world in a more direct and complex way.

Theoretical Development

Components

• Domain of Trigonometric Functions: In trigonometry, the domain is the set of values that an angle can assume. For the trigonometric functions sine, cosine, and tangent, the domain is all real angles (measured in radians).

  • Sin(X): The sine function has as its domain the set of real angles. It always generates an output a real number between -1 and 1. The sine of an angle is the ratio between the length of the opposite leg to the angle and the hypotenuse in a right triangle.

  • Cos(X): The cosine function also has as its domain the set of real angles. Its outputs are real numbers also varying between -1 and 1. The cosine of an angle is the ratio between the length of the adjacent leg and the hypotenuse in a right triangle.

  • Tan(X): The tangent function has as its domain the set of real angles, except for angles that are odd multiples of 90°. Its results can be any real number.

• Range of Trigonometric Functions: In mathematics, the range is the set of values that the function can "produce" when it receives a value from its domain. For sine, cosine, and tangent, the range is the closed interval [-1, 1].

• Period of Trigonometric Functions: For trigonometric functions, the period is the smallest positive value that, when added to the initial angle, results in a new angle with the same function value. In the functions sine, cosine, and tangent, the period is 2π (or 360°).

Key Terms

• Trigonometric Function: It is a function that has as its domain the set of angles and as its range the set of real numbers. The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

• Domain: In the context of trigonometry, the domain is the set of angles for which the function has a corresponding value.

• Range: It is the set of values that the function can assume when the angle belongs to the domain.

• Period: It is the smallest positive value T such that for any angle x, we have that f(x+T) = f(x), where f is a trigonometric function.

Examples and Cases

• Case of the Sine Function: Considering the sine function (sin(x)), if x = π/2, the output of the function will be 1, representing the fact that in the right triangle, the angle π/2 (90°) has the maximum ratio between the opposite leg and the hypotenuse, i.e., the opposite leg has the same length as the hypotenuse. If x = π, the output of the function will be 0, indicating that in the right triangle, the angle π (180°) has the opposite leg with zero length, i.e., the opposite leg coincides with the hypotenuse.

• Case of the Cosine Function: For cosine (cos(x)), if x = 0, the output of the function will be 1, indicating that in the right triangle, the angle 0 has the adjacent leg and the hypotenuse with the same length. If x = π/2, the output will be 0, indicating that the adjacent leg and the hypotenuse are perpendicular, i.e., they have no length in the direction of the other.

• Case of the Tangent Function: In the case of the tangent function (tan(x)), when x = 0 the function is not defined, as it is the point where the sine function is zero. If x = π/4, the output of the function will be 1, as in the right triangle, the angle π/4 (45°) has the maximum ratio between the opposite leg and the adjacent leg, i.e., they have the same length.

Detailed Summary

Relevant Points

• Nature of the Trigonometric Function: A trigonometric function is a function that describes a relationship between the angles of a triangle and the proportions of its side measurements. The main trigonometric functions are sine, cosine, and tangent.

• Domain and Range: For trigonometric functions, the domain is the set of all possible angles (measured in radians), i.e., the input values for the function. The range, on the other hand, is the interval of values that the function can generate when it receives a value from the domain.

• Period of Trigonometric Functions: The period of a function is the smallest value for which the function repeats. In trigonometry, the functions sine, cosine, and tangent have a period of 2π (or 360°). That is, the sine function, for example, will repeat every 2π radians.

Conclusions

• Applications: Trigonometric functions have a wide variety of practical applications, from modeling natural phenomena to calculations in programming and engineering.

• Graphical Interpretation: The analysis of the inputs and outputs of a function, along with its graphical representation, allows for the interpretation of trigonometric concepts in a visual and intuitive way.

• Interconnection of Concepts: The study of trigonometric functions "expands" our understanding of angles and triangles, showing how these concepts are intrinsically related to mathematical calculation.

Exercises

1. Exercise 1: Calculate the value of sin(π/6) and interpret the result.

2. Exercise 2: Determine the angles for which tan(x) = 0 and explain why these angles are not in the domain of the tangent function.

3. Exercise 3: Present the graph of the sine and cosine functions in the interval [0, 2π]. Identify the points of maximum, minimum, and intersection with the x-axis for each of the functions.