Arccsc Calculator (Inverse Cosecant)

csc^-1() =

Degrees:

–

Radians:

–

π radians:

–

Graph of the inverse cosecant

The domain of x is x≤-1 and x≥1.

The range is -π/2≤y<0 or 0<y≤π/2.

Use this calculator to get the result of the inverse cosecant of an entered value. You will be able to get the solution in degrees, radians, and π radians. The allowed domain of x is x≤-1 and x≥1.

Below you can find additional information on how to use the inverse cosecant calculator. In addition, you will be able to explore the definition, graph, and important values of cosecant.

How to use the inverse cosecant calculator?

Step 1: Enter the value of x in the first input box. The entered value must be in the domain x≤-1 and x≥1

Step 2: The result will be displayed in degrees on the right panel.

Step 3: The solution in radians and π radians will be displayed at the bottom.

Difference between degrees, radians, and π radians

The relationship between degrees and radians can be found by considering that we have 360° and 2π radians in one complete revolution. Therefore, we can deduce that 180° is equal to π radians.

On the other hand, we can determine the difference between “radians” and “π radians” by remembering that π has an approximate value of 3.1415… So, 0.5 π radians is equivalent to 1.571 radians since the value of π is included in the second case.

What is inverse cosecant?

Inverse cosecant, also known as arc cosecant, is the inverse function of cosecant. This means that the inverse cosecant reverses the effect produced by the cosecant function.

For example, the cosecant of 30° is equal to 2. Therefore, the inverse cosecant of 2 is equal to 30°.

The inverse cosecant function can be denoted as csc^-1(x) or also as arccsc(x). Since the cosecant function and the sine function are reciprocals, csc^-1(x) is equivalent to sin^-1(1/x).

We can use inverse cosecant to find the angles in a right triangle if we know the ratios of the lengths of its sides. For example, we can find angle A of the following triangle using arccsc(x), where x is equal to c/a.

Why can’t we use values of x between -1 and 1 in the inverse cosecant?

Since inverse cosecant is the inverse function of cosecant, their domains and ranges are swapped. This means that the range of the cosecant is equal to the domain of the inverse cosecant.

Also, there is no value of x that will cause the y-values in the cosecant to be between -1 and 1. Therefore, the x-values of the inverse cosecant cannot be between -1 and 1.

Inverse cosecant graph

We can graph the inverse cosecant if we consider a fixed range. In this calculator, we take a range from -π/2 to π/2 and does not include 0 as shown below:

Inverse cosecant domain

From the graph of inverse cosecant, we can deduce that the values of x can be any real value, excluding values between -1 and 1. Therefore, the domain of inverse cosecant is equal to x≤-1 and x≥1.

Inverse cosecant range

From the graph of the inverse cosecant, we can deduce that the y-values of the function range from -π/2 to π/2 not including 0. Therefore, its range is -π/2≤y< 0 or 0<y ≤π/2.

Table of important inverse cosecant values

Value of x	arccsc(x)(°)	arccsc(x)(rad)
1	90°	$\frac{\pi}{2}$
$\frac{2\sqrt{3}}{3}$	60°	$\frac{\pi}{3}$
$\sqrt{2}$	45°	$\frac{\pi}{4}$
	30°	$\frac{\pi}{6}$
-1	-90°	– $\frac{\pi}{2}$