Exploring the Arctan(x) Function: Properties, Range, and Behavior

arctan(x)

The function arctan(x) represents the inverse tangent function, also denoted as tan^(-1)(x)

The function arctan(x) represents the inverse tangent function, also denoted as tan^(-1)(x). It is the angle whose tangent is equal to x. This function is defined for all real numbers. The output of arctan(x) is an angle given in radian measure.

Let’s understand the properties and behavior of the arctan(x) function:

1. Domain and Range:
– Domain: The domain of arctan(x) is all real numbers, (-∞, ∞).
– Range: The range of arctan(x) is in the interval (-π/2, π/2) or (-90 degrees, 90 degrees), since the tangent function has a horizontal asymptote at these values.

2. Symmetry:
The arctan(x) function is an odd function, meaning that arctan(-x) = -arctan(x). It has rotational symmetry about the origin.

3. Behavior near x = 0:
As x approaches 0 from the left, arctan(x) approaches -π/2. As x approaches 0 from the right, arctan(x) approaches π/2. This means that arctan(0) is equal to 0.

4. Significance of special values:
– arctan(1) = π/4 (45 degrees)
– arctan(0) = 0
– arctan(-1) = -π/4 (-45 degrees)

The arctan(x) function can be used to find the angle whose tangent is equal to a given value. For example, if we want to find the angle whose tangent is 1, we can write it as arctan(1) = π/4 or 45 degrees.

It’s important to note that the arctan(x) function is limited to a specific range of angles. If you need to find angles outside of this range, you may need to use inverse tangent functions with a principal value, such as arctan2(y, x) or atan2(y, x).

Hope this explanation helps in understanding the arctan(x) function and its properties!

More Answers: