37. Trigonometric functions

The trigonometric functions are an essential topic in the complete course of mathematics for the ENEM test. They are a set of mathematical functions that are fundamental to the description of periodic phenomena, such as sound waves and light. In the context of Enem, trigonometric functions are used to solve problems related to triangles, circles and other geometry problems.

Trigonometry has its roots in ancient Greece, where mathematicians studied the relationships between the sides of a right triangle. The concept was then expanded to include the relationships between the angles and sides of any triangle, and later to describe periodic phenomena.

There are six basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (csc). Each of these functions is the ratio of two sides of a right triangle, and each has a specific relationship to the triangle's angles.

The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio between the sine and the cosine of the angle, that is, the ratio between the opposite side and the adjacent side. The cotangent, secant, and cosecant are respectively the inverses of tangent, cosine, and sine.

Trigonometric functions are periodically repeated, which means they have the same value for angles that differ by an integer multiple of a certain angle, called the period of the function. The period of the sine and cosine functions is 2π (or 360°), while the period of the tangent and cotangent functions is π (or 180°).

Trigonometric functions are also harmonic functions, which means that they can be represented as the sum of an infinite series of terms. This property is the basis for Fourier analysis, which is a powerful tool for signal and system analysis.

In addition, trigonometric functions have several important properties that are often used in problem solving. For example, they satisfy various trigonometric identities, which are equations that are true for all values ​​of the variables. Some of the most important trigonometric identities are the Pythagorean identities, which relate the square of the sine and cosine of an angle to unity, and the addition and subtraction identities, which express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles.

In summary, trigonometric functions are an essential topic in the complete course of mathematics for the ENEM test. They are used to solve a wide range of problems, from describing periodic phenomena to solving geometry problems. Therefore, it is important to have a solid understanding of trigonometric functions and their properties to succeed on the ENEM.

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