The idea of cosine is used in the field of geometry. Cosine, in this framework, is the sine of the complement of an arc or an angle, indicates the Royal Spanish Academy (RAE) in its dictionary. The official abbreviation for this trigonometric function is cos, and it is in this form that we find it in equations and on calculators.
It should be noted that the sine is the result of dividing the leg that is opposite an angle and the hypotenuse (in a right triangle, the longest side is the hypotenuse, while the other two –which form the 90º angle– are called legs). The complement, on the other hand, is the angle that, added to another, completes an angle of 90°.
According to abbreviationfinder.org, these concepts belong to the branch of mathematics known as trigonometry, which focuses on the analysis of the so-called trigonometric ratios, among which are the following four, in addition to sine and cosine: tangent, secant, cotangent and cosecant.
In high school, trigonometry is usually included in the last stage of the program, since it is a very complex part and difficult to understand for those who do not have a legitimate taste for numbers. His intervention in the rest of the branches of mathematics is sometimes direct, and other times, indirect; Broadly speaking, we can say that its application takes place whenever it becomes necessary to carry out measurements with a high degree of precision.
Suppose we have a right triangle ABC, with one 90º angle and two 45º angles. By dividing one of the opposite legs at an angle of 45º and the hypotenuse, we will obtain the sine and then we can calculate the cosine.
Another simpler way to calculate the cosine in a right triangle is by dividing the leg adjacent to an acute angle and the hypotenuse. The sine, on the other hand, is obtained by dividing the leg opposite the hypotenuse, while the tangent implies the division of the opposite leg and the adjacent leg. These three functions (cosine, sine, and tangent) are the most relevant to trigonometry.
If a triangle has a hypotenuse of 4 centimeters, an opposite leg of 2 centimeters, and an adjacent leg of 3.4 centimeters, its cosine will be 0.85:
Cosine = Adjacent Leg / Hypotenuse
Cosine = 3.4 / 4
Cosine = 0.85
The secant function, on the other hand, involves the division of 1 by the cosine. In the example above, the secant is 1.17.
The law of cosines, which is also known as the cosine theorem, is a generalization of the well-known Pythagorean theorem. This is the relationship that can be established between one of the sides of a right triangle with the other two and with the cosine of the angle they form.
In a triangle ABC, with the angles α, β, γ and the sides a, b, c (opposite to the previous ones, in respective order), the cosine theorem can be defined as shown in the image: c squared is equal to the sum of a squared and b squared, minus twice the product ab cosγ.
Another way to define cosine is to understand it as:
* an even function: in mathematics, this classification is given to real variable functions taking into account their parity. There are three possibilities: they can be even, odd or have no parity;
* a continuous function: it is a mathematical function in which the points close to the domain carry a series of small variations in their values;
* a transcendental function: it is a function that cannot satisfy a polynomial equation with coefficients that are polynomials (a polynomial is an expression composed of a sum of products of constants and variables with each other).