When it comes to knowing the meaning of the term cotangent, it is necessary, first of all, to discover its etymological origin. In this case, we can state that it is a word that derives from Latin. It is exactly the result of the union of three defined components:
-The prefix «co-«, which can be translated as «together».
-The verb "tangere", which means "to touch".
-The suffix "-nte", which is used to indicate "agent".
Starting from all that, we find the fact that cotangent means "inverse of the tangent of an arc or an angle."
The notion of cotangent refers to the inverse function of the tangent of an arc or an angle . To understand what cotangent is, therefore, we must know what tangent is.
In the context of trigonometry (a specialty of mathematics), the tangent of a right triangle is obtained by dividing the leg opposite an acute angle and the adjacent leg . It should be remembered that the longest side of these triangles is called the hypotenuse , while the other two are called legs .
Returning to the idea of cotangent, we had already mentioned that it is the inverse function of the tangent. Therefore, if the tangent is the quotient between the opposite leg and the adjacent leg, the cotangent is equivalent to the quotient between the adjacent leg and the opposite leg .
In a right triangle whose hypotenuse measures 20 centimeters, its adjacent leg measures 15 centimeters and its opposite leg measures 12 centimeters, we can calculate the cotangent in the following way:
Cotangent = Adjacent leg / Opposite leg
Cotangent = 15 / 12
Cotangent = 1.25
Since the cotangent is the inverse function of the tangent, it can also be obtained by dividing 1 by the tangent . In our example above, the tangent is equal to 0.8 (the result of division between the opposite leg and the adjacent leg). Therefore:
Cotangent = 1 / tangent
Cotangent = 1 / 0.8
Cotangent = 1.25
Within the field of mathematics, and more specifically in the field of trigonometry, the cotangent plays an important role. Specifically, we talk about what the properties of the cotangent function are. And these are none other than continuity, domain, journey, decreasing or period, for example.
Just as the cotangent is the inverse function of the tangent, the cosecant is the inverse of the sine and the secant , the inverse of the cosine .
In the same way, we cannot ignore the existence of what is known as a hyperbolic cotangent. This is another term used in trigonometry in relation to a real number. In that case it is established that it is the inverse of the hyperbolic tangent.
It is represented by coth (x) or by cotgh (x) and on this there is what is called the addition theorem. A theorem that explains the way to synthesize that aforementioned hyperbolic tangent.