Cotangent Calculator

Use this cotangent calculator to find the cotangent of any given degree or radian. All you have to do is put in degrees or radians and the cotangent comes up instantly.

In a right triangle, the ratio of the adjacent side to the opposite side of the acute angle is called the cotangent. Abbreviated as cot.

The formula for cotangent is

cot(θ) = adjacentopposite

For example, there is a right triangle as follows

In this right triangle, the three angles are α, β and γ, where γ is the right angle. The three sides are a, b, and c, where c is the hypotenuse. Therefore, the cotangents of angles α and β are

cot(α) = adjacentopposite = ba

 

cot (β) = oppositeadjacent = ab

Like the tangent, the cotangents of these two angles are reciprocals of each other.

Here comes the question, what is the cotangent of 90°?

Take a look at the picture below!

In a right triangle, when the other angle α is a right angle, that is, the two angles of 90 degrees in the triangle, the remaining angle is 0 degrees. Then, the length of the side adjacent to angle α is 0. Therefore, according to the cotangent formula, it can be concluded that the cotangent of a right angle is 0.

cot(90°) = 0

There are two methods for calculating the cotangent.

In the first method, the lengths of at least two sides of a right triangle are known.

According to the Pythagorean theorem, given the lengths of two sides, the length of the third side can be calculated. Then combined with the cotangent formula, the cotangent of the angle can be easily calculated.

For example, in a right triangle, the side opposite angle α has length 6 and the hypotenuse 10. What is the cotangent of angle α?

First, the adjacent side is computed.

opposite² + adjacent² = hypotenuse²

6² + adjacent² = 10²

36 + adjacent² = 100

adjacent² = 100 – 36

adjacent² = 64

adjacent = 8

Second, calculate the cotangent.

cot(α) = adjacentopposite = 86 = 43

The first step can be omitted if the lengths of the adjacent and opposite sides are known.

In the second method, the degree of the angle is known. In this case, the cotangent can be calculated with the help of a scientific calculator or the cotangent calculator provided on this page. Of course, there is also the most primitive method, refer to the cotangent table (given below) to find the cotangent of the corresponding degree or radian.

In this chapter, we will combine the cotangent curve to summarize the properties of cotangent. Draw the cotangent curve below.

1. Domain – The domain of the cotangent function is all values except kπ. Here, k is an integer.
2. Range – It can be clearly seen that the cotangent curve extends infinitely upwards or downwards, so the cotangent function has no maximum and minimum values. All real numbers are fine.
3. Period – The smallest period is π. cot(θ) = cot(θ + π)
4. Odd function – The graph is symmetric about the origin, cot(θ) = -cot(-θ), so it is an odd function.
5. Decreasing – Monotonically Decreasing in the range.

Furthermore, in different quadrants of the coordinate axis, the cotangent ranges are also different. Cotangent comparisons for the four quadrants are listed below.

1. Cotangent derivative

The derivative of cotangent is equal to the negative of the square of cosecant. Its derivation process is as follows

(cot(θ))’ = (cos(θ)sin(θ))’

= (cos(θ))’sin(θ) – cos(θ)(sin(θ))’sin²(θ)

= -sin(θ) * sin(θ) – cos(θ) * (cos(θ))sin²(θ)

= -sin²(θ) – cos²(θ)sin²(θ)

= –1sin²(θ)

= -csc²(θ)

2. Inverse cotangent

Within a certain range, the inverse cotangent function is arccotangent, denoted as arccot, arcctg or cot^-1. It is a type of inverse trigonometric function. The arccotangent is used to find the angle value from the ratio of the adjacent side to the opposite side.

cot(45°) = 1

cot^-1(0) = arccot (0) = 45°

3. Reciprocal cotangent

The reciprocal of cotangent is tangent, which is equal to the ratio of its opposite to its adjacent side. The abbreviation is tan.

1cot(θ) = tan (θ)

The cotangent calculator is very easy to use, three steps:

First, enter degrees or radians;

Second, choose the type, degrees or radians.

Finally, click Calculate button to get the cotangent answer, or click Reset button to start a new calculation.

All in all, Cotangent Calculator is a simple and handy calculator. It supports not only degrees but also radians, and combinations of π. Everyone is welcome to use it.