Inverse trigonometric functions traditionally cause difficulties for students. The ability to calculate the arctangent of a number may be required in USE problems in planimetry and stereometry. To successfully solve the equation and the problem with the parameter, it is necessary to have an idea of the properties of the arc tangent function.
Definition
The arctangent of x is the number y whose tangent is x. This is a mathematical definition.
The arc tangent function is written as y = arctg x.
More generally: y = Carctg (kx + a).
Calculation
To understand how the inverse trigonometric function of the arc tangent is arranged, you need to first recall how the value of the tangent of a number is determined. Let's consider in more detail.
The tangent x is the ratio of sine x to cosine x. If at least one of these two quantities is known, then the second module can be obtained from the main trigonometric identity:
sin 2 x + cos 2 x = 1.
True, an assessment will be required to reveal the module.
If the number itself is known, and not its trigonometric characteristics, then in most cases it is necessary to approximately estimate the tangent of the number, referring to the Bradis table.
Exceptions are the so-called standard values.
They are presented in the following table:
In addition to those listed above, any values obtained from the data can be considered standard by adding a number of the form ½πk (k is any integer, π = 3.14).
Exactly the same is true for the arctangent: most often, the approximate value can be viewed from the table, but only a few values are known for sure:
In practice, when solving problems in school mathematics, it is customary to give an answer in the form of an expression containing arctangent, and not its approximate estimate. For example, arctg 6, arctg (-¼).
Plotting
Since the tangent can take any values, the domain of the arc tangent function is the entire numerical line. We explain in more detail.
The same tangent corresponds to an infinite number of arguments. For example, not only the tangent of zero is equal to zero, but also the tangent of any number of the form π k, where k is an integer. Therefore, mathematicians agreed to choose values for the arctangent from the interval from −½ π to ½ π. This must be understood as follows. The range of the arc tangent function is the interval (-½ π; ½ π). The ends of the gap are not included, since the tangent is ½n and ½n does not exist.
At the indicated interval, the tangent continuously increases. Therefore, the inverse function of the arc tangent is also continuously increasing on the whole number line, but bounded above and below. As a result of this, it has two horizontal asymptotes: y = −1 π and y = ½ π.
In this case, tg 0 = 0, other points of intersection with the abscissa axis, except (0; 0), the graph cannot have due to increase.
As follows from the parity of the tangent function, arctangent has a similar property.
To build a graph, you should take several points from among the standard values:
The derivative of the function y = arctan x at any point is calculated by the formula:
Note that its derivative is everywhere positive. This is consistent with the earlier conclusion about a continuous increase in function.
The second derivative of the arc tangent becomes 0 at point 0, negative for positive values of the argument, and vice versa.
This means that the graph of the arc tangent function has an inflection point at zero and is convex downward in the interval (-∞; 0] and convex upward in the interval [0; + ∞).