Cosecant

Do you have to calculate the cosecant online of an angle? Use our tool and you can do it either in degrees or in radians. Just type the value of the angle, select the unit it is in and click the calculate button to get the csc(x) value.

If you want to know more about this inverse trigonometric ratio, how it is calculated, its properties and more, read on.



Cosecant formula

Triangle to calculate trigonometric ratios

What is cosecant? It is a trigonometric ratio inverse to sine and that we can express mathematically by the following formula:

Sew-in formula

As you can see, we have multiple ways to calculate the cosecant of an angle. On the one hand we can find the inverse of the sine and, on the other, divide the value of the hypotenuse by the opposite cathetus.

Table of cosecant values

To learn a little more about the cosecant, we are going to see below some of its typical values for usual angles using this table:

GradesRadiansCosecant
0±∞
30ºπ/62
45ºπ/41,4144
60ºπ/31,1547
90ºπ/21
135º3π/41,4144
180ºπ±∞
225º5π/4-1,4144
270º3π/2-1

In this brief review of all the quadrants of a circle, we can see that the cosecant is:

  • Negative in the third and fourth quadrants
  • Positive in the first and second quadrants

This is more clearly represented in the following graph:

Value of the cosecant in each quadrant

Cosecant graph

Now that we know the values of the cosecant of x over the period 2π radians or 360º, we can draw the graph of the function, leaving its representation as follows:

Cosecant graph

As we have already mentioned, you can see the periodicity every 2π radians or 360 degrees.

Features

In addition to being a periodic function, the cosecant has the following characteristics:

  • Domain: all ℝ (the real numbers), except for values that are multiples of a-π. At these values, the function has vertical asymptotes and tends to ±∞ depending on the quadrant we are in.
  • The range is defined on the interval (-∞, -1] U (1, +∞)
  • It does not cut the abscissa axis or the ordinate axis at any point.
  • Since it is symmetric with respect to the origin, it is an even function.
  • Not bounded

Derivative of the cosecant

What is the derivative of the cosecant of x? It is equal to the minus cosecant of x times the cotangent of x. Expressed mathematically it would look like this:

f(x) = cscx → f'(x) = -cscx - cotanx

If instead of x we have the cosecant of a function, we will also have to calculate its derivative and, therefore, it will look like this:

f(u) = cscu → f'(u) = - u' cscsu - cotanu

Integral of the cosecant

If you want to calculate the integral of this function inverse trigonometric, this is the formula you should use:

Integral cosecant

How to calculate the cosecant in Excel

Calculate cosecant in Excel

In case you need calculate the cosecant of an angle in ExcelYou can do this easily regardless of whether the angle is in degrees or radians.

Just type in an empty cell one of the following functions and replace the word "angle" with the number of degrees or radians you need for the calculation.

Angle in degrees:

=CSC((RADIANS(angle)))

Angle in radians

=CSC(angle)

For example, if we want to calculate the cosecant of 60 degreesThe formula that we will write in Excel will be the following:

=CSC((RADIANS(60))))

Cosecant as a function of other trigonometric ratios

Since all trigonometric functions are related to each other, we can express the formula for calculate the cosecant as a function of the sine, cosine or any other. Let's look at each case individually:

Depending on the sine:This is the simplest case since, as we saw at the beginning of the post, the cosecant becomes the inverse of the sine.

Cosecant as a function of sine

Depending on the cosine:

Cosecant as a function of cosine

Depending on the tangent:

Cosecant as a function of tangent

Depending on the secant:

Cosecant as a function of secant

Depending on the cotangent:

Cosecant as a function of the cotangent

All formulas that have the indication (1) means that the result will be positive or negative depending on the quadrant of angle α.

Calculate cosecant in calculator

Cosecant Calculator

Unless you have a very advanced scientific or programmable calculator, you will not find a specific key to calculate the cosecant directly. Therefore, we have to do the calculation in two steps.

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

For example, if we want to calculate the cosecant of 30, we will do the following. First, we obtain the sine by pressing the following key combination:

SIN > 30 > =

With the result we have obtained, we then press the key with the engraving on it x-1 and press the equals (=) button to solve the operation. Here what we have done is to calculate the inverse.

If everything went well, the calculator will display a result equal to 2. If you do not get that, check that you have followed the steps correctly and that you have the calculator set to work in degrees instead of radians.

If you have any type of problem or doubt about the cosecant calculation of an angle, leave us a comment and we will help you as soon as possible. Anyway, we always recommend you to use our online calculator because it is very easy to use and you will avoid any kind of error in the calculations.

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