Arccotangent

The arccotangent is the inverse trigonometric function of the cotangent. It is commonly written as:

$$ y = arccotg \ x $$

Other common notations include arccot, cotan^-1, and cotg^-1.

In simple terms, the arccotangent answers the following question:

Which angle has a given cotangent value?

The answer is an angle measured in radians. On the unit circle, this angle corresponds to a specific arc whose cotangent matches the given number.

For example, the cotangent of an angle of π/2 (90°) is equal to zero:

$$ cotg \ \frac{\pi}{2} = 0 $$

Therefore, the angle whose cotangent is zero is π/2:

$$ arccotg \ 0 = \frac{\pi}{2} $$

Note: Since the arccotangent is the inverse of the cotangent on a chosen interval, applying the arccotangent to a cotangent value returns the original angle:

$$ arccotg ( cotg \ \alpha ) = \alpha $$

Graph of the Arccotangent Function

The arccotangent function takes values in the interval (0, π). Unlike many familiar functions, it is strictly decreasing, meaning that its output becomes smaller as the input increases.

How the Arccotangent Graph Is Obtained

To understand the graph of the arccotangent, it is helpful to start with the cotangent function itself.

The cotangent is a periodic function, which means that its values repeat indefinitely.

Because of this repetition, the cotangent is not one-to-one over its entire domain and therefore cannot have an inverse function on all real numbers.

To make the function invertible, we restrict its domain to the interval (0, π).

Within this interval, the cotangent becomes bijective, establishing a one-to-one correspondence between angles in (0, π) and real numbers.

Note: Other intervals, such as (-π, 0), can also be used. The choice of interval determines the branch of the inverse cotangent function.

Once the domain has been restricted, the cotangent becomes invertible, and its inverse is the arccotangent.

A useful property of inverse functions is that their graphs are mirror images of one another across the line \( y = x \).

For this reason, the graph of the arccotangent can be obtained by reflecting the graph of the cotangent across the line \( y = x \).

Under this reflection, every point \((x,y)\) becomes \((y,x)\), swapping the roles of the horizontal and vertical axes.

The resulting curve is the graph of the arccotangent function.