## Chapter:2

# What are Inverse Trigonometric Functions ?

## Inverse Function

Let f be a one-to-one correspondence from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a)=b. The inverse function of f is denoted by f

^{-1}. Hence, f^{-1}(b)=a when f(a)=b. Inverse trigonometric functions are simply defined as the inverse functions of basic trigonometric functions.## Domains And Ranges Of Inverse Trigonometric Functions

Functions | Domain | Range (Principal Value Branches) |

y = sin ^{-1} x |
[-1,1] | [—ℼ/2 , ℼ/2] |

y = cos ^{-1} x |
[-1,1] | [0, ℼ] |

y = cosec ^{-1} x |
R-(-1,1) | [—ℼ/2 , ℼ/2] - {0} |

y = sec ^{-1} x |
R-(-1,1) | [0, ℼ] - {ℼ/2} |

y = tan ^{-1} x |
R | (—ℼ/2 , ℼ/2) |

y = cot ^{-1} x |
R | (0, ℼ) |

Domain of inverse trigonometry function = Range of trigonometry function.

## Graphs Of Inverse Trigonometric Functions

**(i) Inverse sine function**

In,[—ℼ/2 , ℼ/2], sinx is bijective hence its inverse is y = sin

^{-1}x, x**∈**[-1,1] and y**∈**[—ℼ/2 , ℼ/2].**(ii) Inverse cosine function**

In, [0,ℼ] cosine function is bijective and hence its inverse is
y = cos

^{-1}x, x**∈**[-1,1] and y**∈**[0,**∈**].**(iii) Inverse tangent function**

Tangent function is one-one and onto from (-ℼ/2 ,ℼ/2) to (-∞,∞)
and so in this region the tangent function is invertible i.e.,
y = tan

^{-1}x, x**∈**[-∞,∞] and y**∈**(-ℼ/2 ,ℼ/2)**(iv) Inverse cotangent function**

Cotangent function is one-one and onto from (0,ℼ) to (-∞,∞)
hence cot x is invertible in this region i.e., y = cot

^{-1}x; x**∈**[-∞,∞] and y**∈**(0,ℼ)**(v) Inverse secant function**

A function f : = [0, -ℼ/2) U (-ℼ/2, ℼ] → (-∞,-1] U [1,∞) defined by
f(x) = sec x is one-one and onto and hence it is invertible in this
region, hence y = sec

^{-1}x; x**∈**(-∞,-1] U [1,∞) and y**∈**[0, ℼ/2) U (ℼ/2, ℼ]**(vi) Inverse cosecant function**

f(x) = cosec x is one-one and onto and hence it is invertible in this

region, hence y = cosec-1 x; x**∈**(-∞,-1] U [1,∞) and y **∈**[- ℼ/2, 0) U (0, ℼ/2]