Cotangent

Do you want to calculate cotangent online? With our calculator you can do it for any angle, either in degrees or in radians. After pressing the calculate button you will get the cotg value.

If you also want to learn more about this inverse trigonometric function, in the article we will teach you everything you need to know about it.




Cotangent formula

Triangle to calculate trigonometric ratios

What is the cotangent of an angle? It is a trigonometric function which is calculated by dividing the adjacent leg by the opposite leg. It is also defined as the inverse of the tangent.

Based on the above, we are left with the following formula to calculate the cotangent:

Cotangent formula

Table of cotangent values

Below you will find a table with the usual values that the cotangent takes for typical angles of all quadrants:

GradesRadiansCotangent
0±∞
30ºπ/61,7320
45ºπ/41
60ºπ/30,5773
90ºπ/20
135º3π/4-1
180ºπ±∞
225º5π/41
270º3π/20
315º7π/4-1

If we analyze the results of the table above, we can see that the cotangent value will be:

  • Negative if the angle is in the 2nd or 4th quadrant
  • Positive if the angle is in the first or third quadrant

What we have just told you can be seen in the following graph:

Values of the cotangent depending on the quadrant

Cotangent of 0

What is the cotangent of 0? If you try to solve this answer with the calculator, you will see that you get an error as a result. Why? Let's see it

Earlier we said that the cotangent is the inverse of the tangent, therefore:

tan 0º = 0

Now we calculate the cotg value:

cotg = 1 / tan 0º = 1 / 0 = ±∞

Remember that something between zero equals infinity, but the calculator treats it as an indeterminacy and therefore returns an error. in the calculation.

Why the cotangent of 0 is infinity? To see it more clearly, let's look at the graph of the cotangent function:

Graph of the cotangent

If we look at the value 0 on the x-axis, we can see that the function is not defined but depending on whether we approach from the left or right, its value tends to -∞ or +∞. respectively. We can say that there is an asymptote.

This also occurs for values that are multiples of nπ, where n is an integer.

Features

The main characteristic of the cotangent is that its domain consists of all real numbers (ℝ) except for those angles that are multiples of nπ since, as we have seen in the previous section, at these values the function is not defined and tends to infinity.

Its range is throughout ℝ.

Derivative of the cotangent

The derivative of the cotangent is equal to -1 minus the cotangent to the square of x. We can express this mathematically as follows:

f(x) = cotgx → f'(x) = -csc2x = -1 - cotg2x

Integral of the cotangent

If what we want is to calculate the integral of the cotangent, then we will use this formula:

Integral of the cotangent

How to calculate the cotangent in Excel

Calculate cotangent in Excel

In Excel we can also calculate the cotangent easily with a specific function built into Microsoft's spreadsheet program. To test, open a new spreadsheet and type in an empty cell the following function:

If the angle is in degrees:

 =COT((RADIANS(angle)))

If the angle is in radians:

=COT(angle)

Just change the word "angle" in the formula to the number of degrees or radians you want. For example, if we want calculate the cotangent of 90 we will put the formula in this way:

 =COT((RADIANS(90))))

Cotangent as a function of other trigonometric ratios

Since all trigonometric functions are related to each other, it is possible to express the cotangent as a function of any of them. Below you have all the formulas:

Depending on the sine::

Cotangent as a function of sine

Depending on the cosine:

Cotangent as a function of cosine

Depending on the tangent: this formula is the simplest formula of all since, as we saw at the beginning of the post, by definition, by definition the cotangent is equal to the inverse of the tangent.

Cotangent as a function of tangent

Depending on the secant:

Cotangent as a function of the secant

Depending on the cosecant:

Cotangent as a function of cosecant

All formulas that have the notation (1) on the right side mean that depending on the quadrant to which the angle belongs, the result will be positive or negative.

Calculate cotangent in calculator

Online cotangent calculator

If you want to calculate the cotangent of an angle with calculator. If you do not have a scientific or programmable calculator, you will not find any button to solve this operation. However, there is a method divided into several steps that will allow us to do it.

As we have seen at the beginning of this post, the cotangent is equal to the inverse of the tangent. Therefore, we will divide the calculation of the operation into the following steps:

  1. Calculate the tangent of the angle
  2. Calculate the inverse of the result obtained in the previous step.

For example, let's see what is the cotangent of 30 with the calculator. To do this, the first thing we will do is to calculate the tangent of the angle by pressing the following keys:

TAN > 30 > = =

With the result on the screen, we then have to locate the key that has the symbol x-1 and press the equals (=) button to solve the operation, thus obtaining the inverse which is the value of cotg.

If we have done the steps correctly, the calculator display should show a result equal to 1.73. In case you have obtained something else, check the steps and if you have configured the calculator to work in degrees and not in radians.

In the event that you have any questions or problems related to the cotangent, leave us a comment and we will help you to solve your questions.

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