Inverse trigonometric functions traditionally cause difficulties for schoolchildren. The ability to calculate the arc tangent of a number may be required in USE tasks in planimetry and stereometry. To successfully solve an equation and a problem with a parameter, you need to have an idea about the properties of the arc tangent function.
Definition
The arc tangent of a number x is a number y whose tangent is x. This is the mathematical definition.
The arctangent 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 works, you first need to remember how the value of the tangent of a number is determined. Let's take a closer look.
The tangent of x is the ratio of the sine of x to the cosine of x. If at least one of these two quantities is known, then the modulus of the second can be obtained from the basic trigonometric identity:
sin2 x + cos2 x=1.
Admittedly, an assessment will be required to unlock the module.
Ifthe number itself is known, and not its trigonometric characteristics, then in most cases it is necessary to approximately estimate the tangent of the number by referring to the Bradis table.
Exceptions are the so-called standard values.
They are presented in the following table:
In addition to the above, any values obtained from the data by adding a number of the form ½πк (к - any integer, π=3, 14) can be considered standard.
Exactly the same is true for the arc tangent: most often the approximate value can be seen from the table, but only a few values are known for sure:
In practice, when solving problems of school mathematics, it is customary to give an answer in the form of an expression containing the arc tangent, and not its approximate estimate. For example, arctg 6, arctg (-¼).
Plotting a graph
Since the tangent can take any value, the domain of the arctangent function is the entire number line. Let's 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 arc tangent from the interval from -½ π to ½ π. It must be understood in this way. The range of the arctangent function is the interval (-½ π; ½ π). The ends of the gap are not included, since the tangent -½p and ½p do not exist.
On the specified interval, the tangent is continuouslyincreases. This means that the inverse function of the arc tangent is also continuously increasing on the entire number line, but bounded from above and below. As a result, it has two horizontal asymptotes: y=-½ π and y=½ π.
In this case, tg 0=0, other points of intersection with the abscissa axis, except for (0;0), the graph cannot have due to increase.
As follows from the parity of the tangent function, the arctangent has a similar property.
To build a graph, take several points from among the standard values:
The derivative of the function y=arctg x at any point is calculated by the formula:
Note that its derivative is everywhere positive. This is consistent with the conclusion made earlier about the continuous increase of the function.
The second derivative of the arctangent vanishes at point 0, is negative for positive values of the argument, and vice versa.
This means that the graph of the arctangent function has an inflection point at zero and is convex down on the interval (-∞; 0] and convex upward on the interval [0; +∞).
