#### description

sine, cosine, tangent, cotangent, secant, cosecant function

#### notes

In Class XI, we have studied trigonometric functions, which are defined as follows:

Trigonometric functions | Domain | Range |

sin | R | [-1,1] |

cos | R | [-1.1] |

tan | R- {x:x= (2n+1)π/2, n∈ Z} | R |

cot | R- {x:x= nπ, n∈ Z} | R |

sec | R- {x:x= (2n+1) π/2, n∈ Z} | R- (-1,1) |

cosec | R- {x:x= nπ, n∈ Z} | R- (-1,1) |

We already know that, if f: x→y

f(x)= y and f is bijective,

then there exists g: y→x

g(y)= x

g is called `"f"^-1`

So, with the help of this knowledge we will find the inverse of the trigonometric functions

1)Inverse of sin-

Sin R→[-1,1]

sin x= y

`sin: [-π/2, π/2]→[-1,1]`

`sin^(-1): [-1,1]→[-π/2, π/2],` here `[-π/2, π/2]` is the principle range

`sin^(-1)y= x`

2) Inverse of cos-

cos: R→[-1,1]

cos: [0, π]→[-1,1]

cos (x)=y is a bijective function

`cos^(-1): [-1,1]→[0,π]`, here `[0,π]` is the principle range

`cos^-y=x`

3) Inverse of tan-

`tan: "R"- {x:x= (2n+1)π/2, n∈ "Z"}→"R"`

`tan: [(-π/2), (π/2)]→"R"`

tan x= y is a bijective function

`"tan"^(-1): R→(-pi/2, pi/2)`, where `(-pi/2, pi/2)` is the principle range.

`tan^(-1)y= x`

4) Inverse of cot-

`"cot": "R"- {x:x= n pi, n∈"Z"}→R}`

`cot: (0, pi)→"R"`

cot x= y is a bijective function

`cot ^(-1):" R"→(0, pi)` , where `(0, pi)` is the principle range.

`cot :"R"→ (0, pi)`

`cot^(-1)y= x`

5) Inverse of sec-

`sec: "R"-{x:x= (2n+1) pi/2}→"R"`

`sec: [0, pi]- {pi/2}→"R"- (-1,1)`

sec x= y is bijective function

`sec^-1: "R"- (-1,1) →[0, pi]- {pi/2}`, where `[0, pi]- {pi/2}` is the principle range

6) Inverse of cosec-

`cosec: "R"- {x=n pi, n∈"Z"}→"R"-(-1,1)`

`cosec: [-pi/2, pi/2] -{0}→"R"-(-1,1)`

cosec x= y is bijective function

`cosec^-1: "R"-(-1,1)→[-pi/2, pi/2]-{0}`, where `[-pi/2, pi/2] -{0}` is the principle range.

The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.