Inverse Trigonometric Functions - II

In continuation of the previous post on Inverse Trigonometric Functions , let us define the remaining three functions.

•  The inverse cotangent function , denoted by cot^-1 is the function with domain R,range      (-π/2,0) U (0,π/2], defined by y=cot^-1x  => x = coty The inverse sine function is also called arccotangent , it is denoted by arccot. 
• The inverse secant function , denoted by sec^-1 is the function with domain       (-∞,-1] U [1, ∞), range [0,π/2) U (π/2,π] , defined by y = sec^-1x => x = secy The inverse secant function is also called arcsecant , it is denoted by arcsec. 
• The inverse cosecant function , denoted by cosec^-1 is the function with domain (-∞,-1] U [1, ∞), range [-π/2,0) U (0,π/2] , defined by y = cosec^-1x =>                x = cosecy The inverse cosecant function is also called arccosec , it is denoted by arccosec.

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Criteria for Similarity of Triangles

Inverse Trigonometric Functions - I

For a function to have an inverse, it must be one-to-one. Sometimes, when a function is not one-to-one, one can restrict its domain to make it one-to-one. 

The inverse trigonometric functions are the inverse functions of the trigonometric functions. Because of their periodic nature, the trigonometric functions are not one-to-one. By restricting their domains, we can construct one-to-one functions from them. 

Remembering that the domain of a function and the range of its inverse are the same, we can define the following inverse trigonometric functions : 

• The inverse sine function , denoted by sin^-1 is the function with domain [-1,1] , range [-π/2,π/2] , defined by y = sin^-1x => x = siny .The inverse sine function is also called arcsine , it is denoted by arcsin. 
• The inverse cosine function , denoted by cos^-1 is the function with domain [-1,1] , range [0,π] , defined by y = cos^-1x => x = cosy The inverse cosine function is also called arccosine , it is denoted by arccos. 
• The inverse tangent function , denoted by tan^-1 is the function with domain R , range [-π/2,π/2] , defined by y = tan^-1x => x = tany The inverse tangent function is also called arctangent , it is denoted by arctan.

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Signum Function

In mathematics, the signum function is an odd mathematical function that extracts the sign of a real number. The signum function is defined as

Signum function is also known as sign function.

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Least Integer Function

The function whose value at any number x is the smallest integer greater than of equal to x is called the least integer function. It is denoted by ⌈x⌉ It is also known as ceiling of x. For example ⌈3.578⌉ = 4 , ⌈0.78⌉ = 1 , ⌈-4.64⌉ = - 4.

The graph of the least integer function lies on or above the line y = x , so it provides an integer ceiling for x.

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