Inverse Trigonometric Function

Inverse operations are something we are already familiar with. In addition to being inverse operations, addition and subtraction are also inverse operations, whereas multiplication and division are also inverse operations. Each operation has an inverse that is the polar opposite of the operation in question. A similar distinction can be made between inverse functions of the basic trigonometric functions and the inverse trigonometric functions. There are several names for this function, including arcus functions, anti trigonometric functions, and cyclometric functions. The inverse of the number g is indicated by the symbol ‘g-1‘.

If y = f(y) = sin x, then y = sin-1x is the inverse of y = f(y). Throughout this article, we’ll look at the inverse of trigonometric functions such as the sine and cosine, as well as the tangent and cotangent functions, the secant and cosecant functions.Unlike the other trigonometric functions, the inverse trigonometric functions, such as sine, cosine, tangent, cosecant, secant, and cotangent, execute the opposite action of the other trigonometric functions. We all know that trig functions are very useful when dealing with right angle triangles. It is possible to find the angle measure in a right triangle if two sides of the triangle measure are known. These six important functions are used to do so. The inverse trigonometric function is represented by the convention symbol employing arc-prefixes such as arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x), and arccot(x) .

• The notation sin-1x, cos-1x, tan-1x, and so on denote angles or real numbers whose sine is x, cosine is x, and tangent is x, as long as the values presented are the numerically smallest that are accessible. These are also referred to as arcsin x, arccosine x, and so on.

• If there are two angles with the same numerical value, one of which is positive and the other of which is negative, the positive angle should be chosen.

Domain and range : 

A large number of various angles map to the same value of sin() when the sine function is used. For example, for every integer k,0=sin0=sin(π)=sin(2π)=⋯=sin(kπ). We will restrict our domain before obtaining the inverse sine function in order to avoid the problem of having many values map to the same angle for the inverse sine function in the first place. It is the original function in the domain indicated above, which has been flipped about the line y=xy=x, that is represented by the graphs of the inverse functions. When the graph of an inverse function is flipped about the line y=xy=x, the impact is that the roles of xx and yy are swapped, and this observation holds true for the graph of the function’s inverse as well. For further information, see the wiki page Inverse Trigonometric Graph.

 Properties of Inverse Trigonometric Functions : 

The features of inverse trigonometric functions are determined by the domain and range of the functions being considered. Several features of inverse trigonometric functions are critical to not only solving issues but also gaining a better understanding of the subject at large. To refresh your memory, inverse trigonometric functions are also referred to as “Arc Functions.” They provide the length of arc required to obtain a given value of a trigonometric function for a given value of the trigonometric function. It is defined as the range of values that an inverse function can achieve inside the defined domain of the function when the inverse function is used. Generally speaking, the domain of a function is defined as the collection of all potential independent variables in which the function can be found. Inverse Trigonometric Functions are specified within a specific range of values (interval).

Graphs of Inverse Trigonometric Functions : 

For every trigonometry ratio, there are six inverse trig functions. Six significant trigonometric functions have inverses:

Arcsine

Arccosine

Arctangent

Arccotangent

Arcsecant

Arccosecant

Let’s go over the definitions, formulas, graphs, properties, and solved examples for each of the six major types of inverse trigonometric functions. 

Arcsine Function : 

The arcsine function is the inverse of the sine function, which is represented by the symbol sin-1x. Using this graph, you can see how it is depicted:

Arccosine Function :

The arccosine function is the inverse of the cosine function, which is represented by the symbol cos-1x. Using this graph, you can see how it is depicted:

Arctangent Function : 

tan-1x is the arctangent function, which is the inverse function of the tangent function.Using this graph, you can see how it is depicted:

Arccotangent (Arccot) Function : 

The arccotangent function, indicated by cot-1x, is the inverse of the cotangent function, which is denoted by cot. Using this graph, you can see how it is depicted:

Arcsecant Function : 

The arcsecant function is the inverse of the secant function, which is symbolised by sec-1x in the notation of mathematics.Using this graph, you can see how it is depicted:

 Arccosecant Function 

This function is the inverse of the cosecant function, which is represented by the symbol cosec-1x. Using this graph, you can see how it is depicted:

Conclusion : 

Inverse operations are something we are already familiar with. In addition to being inverse operations, addition and subtraction are also inverse operations, whereas multiplication and division are also inverse operations. Each operation has an inverse that is the polar opposite of the operation in question.The features of inverse trigonometric functions are determined by the domain and range of the functions being considered. Several features of inverse trigonometric functions are critical to not only solving issues but also gaining a better understanding of the subject at large.