This results in the graph of the inverse function of the cotangent, known as the arccotangent. Within the interval (0, π), the cotangent is an invertible function. There are infinite intervals where the cotangent function is bijective, such as (-π, 0) or (π, 2π). Generally, the interval (0, is swing trading safer than day trading is it less risky π) is used, but other intervals can be chosen if needed. However, by restricting its domain to the interval (0, π), the cotangent becomes bijective. The table below lists the primary angles of the cotangent.

Alternative names of cotangent are cotan and cotangent x. Let us learn more about cotangent by learning its definition, cot x formula, its domain, range, graph, derivative, and integral. Also, we will see what are the values of cotangent on a unit circle. The cotangent function (cot(x)) is a trigonometric function defined as the ratio of the adjacent side and opposite side of a right-angled triangle. Since the cotangent is a periodic function with a period of π, it can be studied within the interval (0, π). In this interval, the cotangent is a continuous, monotonic, and decreasing function.

Thus, the graph of the cotangent function looks like this. The cotangent function is periodic with a period of π and is undefined at odd multiples of π/2 (i.e., cot(π/2), cot(3π/2), cot(5π/2), etc.). The cotangent function has vertical asymptotes at these values of x.

• However, by restricting its domain to the interval (0, π), the cotangent becomes bijective.
• Since the cotangent function is NOT defined for integer multiples of π, there are vertical asymptotes at all multiples of π in the graph of cotangent.
• Generally, the interval (0, π) is used, but other intervals can be chosen if needed.

Tangent Function : f(x) = tan (x)

The cotangent is undefined at angles where the y-coordinate (sine) is zero. At these points, division by zero occurs, resulting in an undefined ratio. Therefore, the cotangent is not defined at angles where the sine is zero. The domain of the cotangent function includes all real numbers, except for integer multiples of π.

Properties of Trigonometric Functions

To obtain the graph of the inverse function, rotate the graph 90° counterclockwise. Understanding these properties helps solve equations and simplify expressions.

Just like other trigonometric ratios, the cotangent formula is also defined as the ratio of the sides of a right-angled triangle. The cot x formula is equal to the ratio of the base and perpendicular of a right-angled triangle. Here are 6 basic trigonometric functions and their abbreviations.

What is Cotangent in Trigonometry?

Since both sine and cosine functions have a period of 2π, when we observe cotangent, it effectively cancels out some of this periodicity. As we look at the behavior of cot(x) over the interval of 0 to 2π, we’ll see that it actually completes a full cycle and returns to its initial value just after π. It is, in fact, one of the reciprocal trigonometric ratios csc, sec, and cot. It is usually denoted as “cot x”, where x is the angle between the base and hypotenuse of a right-angled triangle.

Integral of Cotangent

• There are infinite intervals where the cotangent function is bijective, such as (-π, 0) or (π, 2π).
• Here are 6 basic trigonometric functions and their abbreviations.
• The periodicity identities of trigonometric functions tell us that shifting the graph of a trigonometric function by a certain amount results in the same function.
• Since both sine and cosine functions have a period of 2π, when we observe cotangent, it effectively cancels out some of this periodicity.
• Here, segment OA represents the cosine, and segment OB represents the sine of the angle α that defines point P.

Also, we will see the process of graphing it in its domain. The graph of cot(x) is symmetric about the x-axis and has many vertical asymptotes. It oscillates between positive and negative values as x increases or decreases. The cotangent is a periodic function that repeats every π (or 180°). We can also find the above properties from the graphs of the respective trigonometric functions.

Trigonometric functions are the simplest examples of periodic functions, as they repeat themselves due to their interpretation on the unit circle. Where cos(x) is the cosine function and sin(x) is the sine function. The inverse function of the cotangent is the arccotangent. Thus, the cotangent can also be expressed as the reciprocal of the tangent. Hence, the value of the cotangent ranges from -∞ to +∞. By definition, the cotangent is the ratio between the x-coordinate OA and the y-coordinate OB of point P.

Practice Questions on Cotangent

Here, segment OA represents the cosine, and segment OB represents the sine of the angle α that defines point P. Each trigonometric function has distinct restrictions on its inputs (domain) and outputs (range), primarily due to its periodic and geometric nature. So basically, if we know the value of the function from \(0\) to \(2\pi\) for the first 3 functions, we can find the value of the function at any value. More clearly, we can think of the functions as the values of a unit circle.

The term “cotangent” was first introduced by the English mathematician Edmund Gunter in the 17th century. This is obtained by extending the radius OP until it intersects at point K with a line parallel to the x-axis, which is tangent to the unit circle at CK. From a geometric perspective, the cotangent corresponds to the segment CK.

Since the cotangent function is NOT defined for integer multiples of π, there are vertical asymptotes at all multiples of π in the graph of cotangent. Also, from the unit circle (in one of the previous sections), we can see that cotangent is 0 at all odd multiples of π/2. Also, from the unit circle, we can see that in an interval say (0, π), the values of cot decrease as the angles increase.