Trigonometric Functions - Basics

INTRODUCTION TO TRIGONOMETRIC FUNCTIONS

Trigonometric functions, also known as goniometric functions, are fundamental tools in mathematics that allow for the description of relationships between sides and angles in triangles, especially in right-angled triangles. These functions are crucial and form the basis for understanding trigonometry.

BASIC TRIGONOMETRIC FUNCTIONS

The basic trigonometric functions include:

• SINE (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
• COSINE (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
• TANGENT (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, which is also equal to sine divided by cosine (sin/cos).

KEY PROPERTIES OF TRIGONOMETRIC FUNCTIONS

Trigonometric functions have several important properties:

• PERIODICITY: Sine and cosine are periodic functions with a period of 2π, while tangent and cotangent have a period of π.
• SYMMETRY: They exhibit specific symmetries on the unit circle, such as being odd or even functions, and symmetries with respect to the axes and the origin.
• BOUNDEDNESS: Sine and cosine are bounded functions; their values range between -1 and 1. Tangent and cotangent are not bounded.

USES OF TRIGONOMETRIC FUNCTIONS

Trigonometric functions are useful in many areas of mathematics and science:

• GEOMETRY AND CARTOGRAPHY: They are used to calculate distances, angles, and areas of geometric shapes, as well as for navigation and location services.
• PHYSICS: Trigonometric functions are essential for analyzing periodic phenomena, such as waves and oscillations, and for solving problems related to vectors and forces.
• ENGINEERING: They are used in the design and analysis of structures, electrical circuits, and signal processing.
• ASTRONOMY: They enable the calculation of the positions and movements of celestial bodies.

CONCLUSION

These functions are fundamental elements of trigonometry, allowing for precise analysis and understanding of geometric and physical phenomena. Their versatile utility and connection with various mathematical identities classify them among the basic tools in mathematics, natural sciences, and technical disciplines. The study and use of trigonometric functions are key to the development of new technologies, scientific discoveries, and the improvement of engineering solutions, which confirms their immense value. Understanding trigonometric functions opens the door to a wide range of applications, from solving practical problems to exploring fundamental natural laws.